Well, Wizards released an entire web enhancement devoted to cats as an April Fools article, which has a lot of cool options for cats. It does have at least one “playable” cat (Cheshire Cat, to be exact), but it’s not really a very playable option because it has LA +3. If it lost the DR, toned down the ability score bonuses a bit, it could possibly pass as a more playable LA +1.
The more typical answer to the desire to play as a cat is to use the tibbit race from Dragon Compendium. These shapeshifters can take on the form of a housecat, and are an LA +0 race (i.e. in line with the regular races). In their regular form, they are Small, have −2 Strength but +2 Dexterity, and Darkvision out to 60 ft.
Catfolk are in Races of the Wild, but they’re fairly disappointing. They have LA +1, and get very little to show for it, sadly.
Shifters from Eberron Campaign Setting or Races of Eberron are quasi-lycanthropes that can “shift” or become more bestial, with fangs or claws or what have you. Could be somewhat feline, but mostly it’s just humanoid with claws etc. They are LA +0, but tend to be fairly underwhelming even for an LA +0 race: while shifting, they’re pretty typical, but that means when they’re not shifting, they’re sub-par.
Both tibbits and catfolk can probably qualify as “feline” for the purposes of the feats and similar in the Fabulous Cats! article. Shifters could if you let them, I suppose.
As for homebrew, bhu has a fair amount of kitty-related homebrew, and he’s a fairly noted ’brewer. I haven’t personally read any of his cat stuff, though.
The premise of your question is somewhat incorrect. Level 20 is a standard progression limit only in 3e and 5e.
In First Edition AD&D, there is no level limit. Specific class advancement tables describe advancement from anywhere from 29 levels (cleric) to 9 levels (fighter) but only for purposes of showing how high certain abilities can go, they all note you can go on up infinitely from there. The big differentiation is "name level," which is usually in the 9-10 range, where the character stops getting as large level advances in hp and starts focusing on building kingdoms and whatnot.
In BECMI D&D (Red Box Basic), you can go up to level 36, and there are level breaks from Basic (up to level 3), Expert (up to level 14), Companion (up to level 25), Masters (up to level 36), and Immortal (past that, cashing in XP for power) with differences in those levels of play.
In Second Edition AD&D, advancement is described on convenient charts up to level 20 but there is no limit, with a breakpoint at level 9-10 where you stop getting full hit points with each level. It has a section in the DMG about how play gets harder to be satisfying at higher levels and that you probably need to shift campaign styles. In terms of playstyle recommendation past 20 it says
Consummate skill and creativity are required to construct adventures for extremely powerful characters (at least adventures that consist of more than just throwing bigger and bigger monsters at the nearly unbeatable party). Very high level player characters have so few limitations that every threat must be directed against the same weaknesses. And there are only so many times a DM can kidnap friends and family, steal spell books, or exile powerful lords before it becomes old hat.
It then recommends retirement as an endgame.
In Third Edition D&D, advancement is described up through level 20, with levels past that described in "an upcoming rulebook." It was lightly treated in the DMG but then more fully in the Epic Level Handbook in the 3.5e days. 3.5e and Pathfinder are lightly changed derivatives of 3e by design and so aren't really different editions with different ideas driving them as far as this goes. The Epic Level Handbook describes its intention, which is to change playstyle from the level 1-20 model to being legendary, allowing PCs to "wield powers that other characters (even 20th-level ones) can only dream about." It notes that PCs may have had the thrills of running nations and political machinations come and go and this is their gateway to discovering the secrets of the universe, plugging into the primal cosmic battles, etc.
You may also want to review the 3.5e DMG's discussion on epic characters and why attack and save bonuses cap out at 20 on p.207. (Summary: too many attacks causes slog and too much disparity between faster and slower save and BAB progressions causes balance issues). Also on p.210 they explain that many classes have been balanced assuming that 1-20 progression and that balancing classes for infinite progression is way harder.
In Fourth Edition D&D, the level limit is 30. There is no implied "soft cap" at 20. Play is grouped into rough "tiers" from the heroic (1-10) to paragon (11-20) to epic (21-30), but it is a continuum.
In Fifth Edition D&D, the game describes four tiers of play (1-4, 5-10, 11-16, 17-20) with "epic boons" available after level 20.
From this, you can take away several things.
The nature of play changes with level. Kicking down the door and killing something works well as a low level adventure and less well as a high level adventure, due to both repetition and the powers and abilities available to higher level PCs and foes, so shifting to more political or grander-scale adventures becomes desirable. Some editions formalize this with tiers, others say "low level, high level, and very high level", etc. What level that is varies by the specific D&D edition and its core rules.
There are a variety of "soft caps" and "hard caps" across the editions - tier boundaries, name level, etc. Only in 3e and 5e is 20 specifically a meaningful number that one might describe as a levelling cap (with later progression options). These numbers are not based on some arcane math but on when the designers feel like gameplay breaks down under its prior level paradigm. 4e is reusing the "epic" term but there is a continuum from 1-30 where epic can't be considered a meaningful cap, even a soft one, it's a breakpoint like the one at level 10. It's basically just using previous edition words for that level band to comfort people.
"Epic Level" play is a 3e thing based on a very specific product and terminology coined for 3e. Most references you've seen to "post-20 play" and "epic" are just an outgrowth of 3e play specifically. You are seeing something "across all of D&D" which isn't really across all of D&D.
Since versions of D&D mostly share certain rule similarities, the breakpoints of power - mostly cemented by what spells become available (fly, teleport, wish, etc. change the dynamics of the game by their availability) tend to be in around the same spots. So short of devising new things (tenth level spells, epic powers, etc.) versions of D&D that use the traditional spell advancement of "a new spell level every couple character levels" cap out spell power right before 20, where then it becomes a game of "more, but not really different" without additional rules that are pointless to include in core books where 99% of people don't ever get up to level 20 anyway. But this means that the around-level-20 breakpoint isn't really deliberately designed, it's more of an inevitable endgame of the spell system, unless you deviate from it (as 4e did). Even in BECMI, the Master rules (level 26-36) are only 32 pages long and are basically some new spells and then siege engine rules. When I played Basic no one ever went past Companion because the game got pretty weird and uninspired there.
Given a class-and-level system of D&D's kind, and the kind of Vancian casting powers traditionally available at levels around 5 (fly, fireball), 10 (teleport, raise dead), 18 (wish, miracle) then you get a similar need to change playstyles at those milestones, with 1-5 being your gritty stuff, 6-10 being (super)heroic, above that needing to change more to political and larger scale concerns to keep challenge and interest, and around 20 becoming a point of diminishing returns where you need to do something different to maintain challenge and interest given how spells etc. cap out there. BECMI Master pushed this past to 36 and got boring for that whole range; 4e went to 30 by discarding the Vancian tradition other editions share.
So it is incorrect to say that 20 is a soft cap across most editions, but this is the reason behind it in 3e/5e and the other "caps" and "breakpoints" and "tiers" in other editions in general. It's an emergent condition of the kind of ruleset D&D is and its historical trappings (Vancian magic being the most important) driving a change in playstyles at certain power inflection points. The designers explicitly talk about this in each edition's books regarding high/various level play.
If it needs to be stated more simply, 20 is not a magic number, it's just when having 9th level spells gets old.
Best Answer
D&D 3.5e
D&D “v.3.5 revised edition” is by-far the edition I know best, so here’s what can be found in that edition.
Ooze-like officially-playable races
The ghaunadan are a race of evil ooze-like shapechangers devoted to Ghaunadaur, and in 3.5e they were technically playable, though in practice I don’t think it would have been plausible.1 However, they were not technically oozes—their originally incarnation in Monsters of Faerûn had them as “shapechangers” which isn’t a thing that can stand on its own in 3e (it’s a “subtype”), and the errata to Player’s Guide to Faerûn (which officially was the last word on updating a lot of things to the revised edition) states they are aberrations (the same type used by illithids, beholders, and the like).
Also in Faerûn, we have the Underdark book2 and its slyth race. The slyth are regular humanoids, but they do have the ability to turn into an ooze-like “amorphous form” for up to 10 × level minutes per day, broken up as the slyth sees fit. That form is pretty much solely for infiltration, though, since the slyth isn’t really allowed to do much while using it except move around (including through incredibly tight spaces). Anyway, it is officially, technically, playable, though again I suspect most would find the experience frustrating.3
Intelligent oozes, including gelatinous cubes—not officially playable
One of the hard, practical stops on playing any monster is Intelligence—even if you ignore the rules saying you’re not allowed to play one, you can’t really turn a mindless creature into a player character. Most oozes—including the gelatinous cube—are mindless, but D&D 3.5e does provide ways around that.
For example, there’s the awaken ooze spell (Dragon vol. 304), or the sentry ooze template (Dungeonscape), though the latter only provides “animal-like” Int 2. There are other templates that are not specific to oozes, but still legally applied to one, that provide more Intelligence. Ultimately, however, none of these rules officially make the cube playable.
Wizards used those same ideas in its own “Elite Opponents” article that includes a gelatinous cube monk, but it’s not playable either—monsters could take class levels in this edition. It gets around the gelatinous cube’s lack of Intelligence by using the fiendish creature template. They did similar things with other oozes in another “Elite Opponents” article.
There are also various oozes that are intelligent by default, mostly in Monster Manual II and III. These aren’t playable either.
Ooze-related class features, including becoming an ooze
Clerics could have the ooze domain (Fiendish Codex I: Hordes of the Abyss) and/or the slime domain (Player’s Guide to Faerûn), allowing them to rebuke and command oozes they found while adventuring. Such a cleric could build up an ooze army. Both domains offer this same feature, but have different associated spells and in any event you could take both to get more uses per day. And maybe you’d be able to use the two command abilities separately, effectively allowing you to command twice as many oozes? I don’t believe there’s an official rule on that.
Sorcerers and wizards can try to do something similar by using the 6th-level spell ooze puppet (Spell Compendium). That doesn’t appear to have any cap on how many puppets you can have at once, beyond your available spell slots. Considering that each casting lasts a minimum of 11 days, that could be a lot. On the other hand, as an 11th-level character, many of your enemies are probably not going to be too scared by most oozes, even in large numbers.
Speaking of spells, there are several other relevant and related spells:
Mold touch (Player’s Guide to Faerûn) is a 3rd-level spell available through the Initiate of Nature feat. It summons brown mold on the spot you touch, which can be another creature. Brown mold isn’t a creature in 3.5e, so it doesn’t have a type, and might arguably be more of a fungi thing than a slime thing. Then again, there’s overlap there.
Engulfing terror (Drow of the Underdark4) summons a gelatinous ooze as a 3rd-level druid, sorcerer, or wizard spell.
Amorphous form (Spell Compendium) is a 3rd-level spell for assassins, sorcerers, and wizards that allows you to take on many ooze traits temporarily. Doesn’t technically make you an ooze though.
Touch of Juiblex (Book of Vile Darkness) is an absurd 3rd-level “corrupt” spell that turns its target into green slime over 4 rounds. A corrupt spell can be used by anybody willing to pay the price, in this case taking 1d6 Strength damage. Anyway, during those 4 rounds, you need remove curse, polymorph other (a 3e spell that was removed in the 3.5e revision), heal, greater restoration, limited wish, miracle, or wish to undo it. After the 4 rounds, there doesn’t seem to be anything you can do about it, except I guess destroying the green slime and casting true resurrection. Maybe resurrection on the slime itself? Regardless, those are high-level spells that a lot of targets are not going to have handy, particularly not at 5th level, making this a monstrously effective tactic. I’d ban it, to be honest.
Mantle of the Slime Lord (Champions of Ruin) is a 7th-level spell available via the Initiate of Ghaunadaur feat. It gives you some ooze traits and makes mindless oozes ignore you.
Simbul’s skeletal deliquescence (Magic of Faerûn) is an 8th-level sorcerer or wizard spell that “is normally used as a punishment or to disable opponents without killing them,” by turning their bones to “mush” and making them “oozelike.” It actually does provide the target some of the benefits of being an ooze, but since it doesn’t provide the target with any ability to move without their bones, this is small comfort.
Wall of ooze (Book of Vile Darkness) is a 5th-level spell for clerics, sorcerers, and wizards that does what it says. The wall isn’t a creature, per se, nor is it mobile or anything, but anyone who touches it does take acid damage and risk paralysis. If they are paralyzed, the wall tries to digest them, and adds their hp to its own if successful.
Yochlol blessing (Drow of the Underdark4) is a 4th-level cleric spell that turns someone into a yochlol. Yochlols are kinda oozy.
D&D 3.5e was also well-known for its panoply of “prestige classes,” classes you couldn’t start out in but had to multiclass into after meeting the class’s requirements. Several are relevant here:
Slime lord (Player’s Guide to Faerûn), another thing devoted to Ghaunadaur, becomes very ooze-like themselves, gaining most of the features associated with oozes over 10 levels. They do not, however, technically change to the ooze type.
Thralls of Juiblex (Book of Vile Darkness) get a lot of features similar to the slime lord—many ooze traits, but no official ooze type.
Oozemasters (Masters of the Wild) actually do turn into oozes, technically and officially. Weirdly, though, they don’t really get any kind of “mastery” over oozes—even if you were a slime-domain cleric who multiclasses into oozemaster, it won’t progress your ability to command slimes, so you’ll be really limited on that front. It does progress your spellcasting at most levels,5 though, so I guess you could use spells to summon oozes.
Spell sovereigns (Dragon vol. 357) are all about living spells, which are just that, spells turned into creatures. Living spells are oozes under 3.5e rules, so technically this class is about (one specific kind of) oozes. But not oozes in general, and in any event the spell sovereign only controls living spells; they don’t become one. On the other hand, it’s also definitely the most powerful way to specialize in oozes, since living spells are incredibly flexible and the spell sovereign gets one as a familiar, which opens up all kinds of familiar-related class features, feats, and spells. That includes options like the Extra Familiar feat (Dragon vol. 280) to get more than one, and a changeling wizard’s morphic familiar feature (Races of Eberron), to swap around which spells each of your familiars is. On top of that, it advances your existing spellcasting at most levels.5 (Credit here to @forrestfire, who is playing a character similar to what I’ve described here in a game we’re in and worked out this approach.)
Master of many forms (Complete Adventurer) is a class devoted to, well, mastering many forms, expanding on a druid’s wild shape and getting into all kinds of creatures—including oozes at 8th level. They can actually become an ooze, but only temporarily. It lasts a long time and they can use it a lot because they’re heavily specialized in it, so they probably could stay in ooze form all day, every day if they really wanted, but it’s not what the class is for.
D&D 3.5e used a “Level Adjustment” system, where a character of a “powerful” race like ghaunadan counts as a higher level than they actually are. A ghaunadan’s is +5, so a 1st-level ghaunadan is supposed to count as a 6th-level character. The benefits of being a ghaunadan do not—remotely—compare with the benefits of five class levels, making any ghaunadan character feel very much “behind” their nominal peers—in fact, by comparison, the ghaunadan will most likely feel utterly crippled. A 1st-level ghaunadan is simply not up for dealing with 6th-level threats.
Not to be confused with Drow of the Underdark, which is a setting-agnostic supplement with similar themes from the same edition.
Like ghaunadans, slyth had an LA. It was only +2 instead of +5, but again nothing about the slyth was even remotely worth two class levels. It might just barely be something some masochist was willing to put themselves through, but I cannot more strongly recommend against it.
Not to be confused with just Underdark, the Forgotten Realms supplement mentioned earlier.
Note, however, that missing even one level of spellcasting progression is widely considered near-crippling in D&D 3.5e. Spells are just that powerful. On the other hand, spellcasters are so powerful they can afford to cripple themselves a little.
D&D 4e and 5e
D&D 4e or 5e don’t have any playable oozes. Both editions have a much, much more limited selection of playable races, and neither provide an official way to play as monsters, so it’s fairly easy to check that none are oozes or even ooze-like.
4e playable races
Human, Half-Elf, Halfling, Dwarf, Elf, Eladrin, Dragonborn, Tiefling, Deva, Half-Orc, Longtooth Shifter, Razorclaw Shifter, Goliath, Gnome, Minotaur, Githzerai, Shardmind, Wilden, Drow, Genasi, Kalashtar, Changeling, Warforged, Mul, Thri-Kreen, Gnoll, Revenant, Shadar-Kai.
5e playable races
Dragonborn, Dwarf, Elf, Gnome, Half-Elf, Halfling, Half-Orc, Human, Tiefling, Changeling, Kalashtar, Shifter, Warforged, Aarakocra, Genasi, Goliath, Centaur, Loxodon, Minotaur, Simic Hybrid, Vedalken, Aetherborn, Aven, Khenra, Kor, Merfolk, Naga, Siren, Vampire, Aasimar, Bugbear, Firbolg, Goblin, Hobgoblin, Kenku, Kobold, Lizardfolk, Orc, Tabaxi, Triton, Yuan-ti Pureblood, Gith, Locathah, Tortle, Verdan
Earlier editions
I have absolutely no idea; you’ll need an answer from someone well-versed in those systems for those.
Pathfinder third-party material
This isn’t really D&D, since it wasn’t published by D&D’s owners or even for D&D itself, and for that matter, it’s not even really Pathfinder, since it wasn’t published by Paizo. But it’s also exactly what you’re looking for, and Pathfinder is kinda-sorta D&D.
There is a “pay what you want” third-party publication by Dreamscarred Press, April Augmented, that includes a playable gelatinous cube. Converting Pathfinder material to D&D 3.5e material (which it’s based on) is not hard (but does also require some effort, you can’t just use it as-is).
Disclaimer: That’s April Fools’ material. DSP took pride in even their joke material, and all of it is intended to be usable even in serious games, but it’s worth noting.
Disclaimer the second: I contributed to that book. I didn’t work on the playable gelatinous cube, and it was a volunteer gig in the first place, but you should be aware that I have a vested interest in this publication.