For the Core Rulebook alone, this is actually not to hard to figure out:
Race
None of the core races has a base speed greater than 30ft. Human is a great choice for the bonus feat.
Feats
Being Human, we have 11 feats at 19th or 20th level. These are quite simple to assign: We'll put one of those into Run and the other 10 into Fleet, which explicitly stacks with itself, for a flat 50ft. untyped bonus to our land speed.
Class(es)
Among the 11 core classes, Barbarian (Fast Movement, land speed, untyped), Druid (Wild Shape, replaces base speed), Cleric (Travel Domain, all speeds untyped), and the Monk (Fast Movement, land speed, enhancement bonus) have class features for increasing their speed that are not spells.
Land speed
Land speed is quite difficult to achieve, but note that land speed is not limited by any of the Beast Shape forms, thus requiring fewer druid levels. A 4th level druid can transform into a Deinonychus with 60 ft. base land speed.
It does get a little tricky with the Polymorph rules:
While under the effects of a polymorph spell, you lose [...] any [...] movement types possessed by your original form. You also lose any class features that depend upon form [...] The GM is the final arbiter [...]
But considering that an awakened deinonychus monk/barbarian would have both Fast Movements, I'll assume our Monk/Barbarian/Druid can keep both them in Wild Shape.
We don't need more than 1 level of Barbarian for Fast Movement, or more than one level of Cleric for Travel Domain. We'll put 12 levels in Monk, for a 40ft. enhancement bonus to speed (level 13 and 14 do not increase our speed). The last 2 levels can be put in druid to increase the duration of our wild shape from 4 to 12 hours.
Our Human Barbarian 1/Cleric 1/Druid 6/Monk 12 now has a land speed of 170 ft. in dinosaur form (12 hours a day), which is quintupled when using the Run action, for a total of 850 ft. of movement per round, which is 96 miles per hour.
Fly speed
Flying is actually easier: Let's look at transforming into an Air Elemental using the Elemental Body IV variant:
Air elemental: As elemental body I except that you gain a +4 size bonus to your Strength, +6 size bonus to your Dexterity, and a +4 natural armor bonus. You also gain fly 120 feet (perfect).
This is available at 12th, leaving us with 8 spare levels. Taking 2 levels of Barbarian, for the Swift Foot rage power, giving a 5ft. enhancement bonus to all speeds, and 1 level of Cleric for Travel Domain's 10 ft. bonus brings us to a total of 135 ft. while raging (130 otherwise), before feats.
Over all, our Cleric 1/Barbarian 2/Druid 12/X 5 ends up with a fly speed of 185 ft., quintupled when "running", for 925 ft. movement per round, which is about 105 miles per hour.
The short answer:
AC 0-5: Human Great Weapon Fighter with the Great Weapon Master feat wielding a Greatsword
AC 6-20: Raging Human Barbarian with the Polearm Master feat wielding a Polearm
AC 21-26: Half-Orc Two-Weapon Fighter
AC 27+: Human Great Weapon Fighter with the Great Weapon Master feat wielding a Greatsword, part II: the GWF returns.
The significantly longer answer is that here are some calculations I did for the options that seemed obvious to me. Interestingly enough, the Barbarian didn't make the top of the list in terms of pure damage, but it did make the top of the list once we added AC into the mix.
You can also add Caltrops to any of these builds (and you can afford a lot of caltrops) to add up to 1 dmg/r.
Other notes:
- The only source material used for this was the PHB, as I do not have access to all of the adventure paths and supplements
- Magic users need not apply: Cantrips are almost universally low-damage, and they never get to benefit from base stats to get those sweet, sweet flat bonus damage numbers. All other spells are too consumable to be considered, so they didn't make it to the number crunching stage.
- Rogues were briefly considered, but sneak attack just isn't reliable enough when you can't count on nearby allies or advantage.
- No cheese! I tried to stick to a purist approach here. There may be strategies that your GM approves like this but most reasonably GMs would ban that kind of build from their tables for being too OP. This list fits nicely within very vanilla RAW and RAI.
- I gave all characters rolls of 18 for their stats, then added any racial strength bonus that applied. Point-buy only allows you to get to 16/17 instead of 19/20, and will lower your dmg/r across the board.
- All damage is given in expected values, done by hand. It's possible I failed my Intelligence(Statistics) check, so if you see something say something
- Humans used are Variant Humans, to take advantage of the bonus feat. Half-Orc is used otherwise as they're the only race I could find that can possibly add damage to their attacks through the Savagery feature, as well as giving a handy +2 Str.
- No magic weapons are used, as they are out of the budget of level 1 characters even if they sell all of their possessions, max out their starting gold rolls, and take the Noble background
- Using the point-buy system the half-orc takes a bigger hit than humans to both his to-hit values and his damage (the +2 doesn't get him to a higher Ability Modifier tier). This results in the Barbarian Polearm Master reigning supreme all the way to 24 AC, and the GWF picks up the slack at 25+, cutting the half-orc out of our equation completely.
- You want numbers? Here's a Google Sheet with some numbers.
Without further ado, here is my list of damage-optimized level 1 characters (please refer to the above sheet for numbers with AC factored in).
Human Great Weapon Fighter with Great Weapon Master feat (22.75 dmg/r, +1 to hit)
- +4 str bonus
- +10 feat bonus (-5 to hit)
- 2d6 (reroll 1,2) greatsword attack (22.333333 dmg/r)
- 5% crit (.41666666 dmg)
- Total damage on maximum Crit: 14+12+12 = 38 dmg
Human Barbarian with Polearm Master feat (20.3 dmg/r, +6 to hit)
- +4 str bonus
- +2 rag bonus
- 1d10 polearm attack (11.5 dmg)
- 1d4 bonus attack (8.5 dmg)
- 5% crit (0.4 dmg)
- Total damage on maximum Crit w/ both attacks: (6 + 10 + 10) + (6 + 4 + 4) = 38 dmg
Half-Orc Two Weapon Fighter (17.4 dmg/r, +7 to hit)
- +5 str bonus
- 1d6 shortsword attack (8.5 dmg)
- 1d6 shortsword attack (8.5 dmg)
- 5% crit including extra damage die from Savagery (0.7 dmg)
- Total damage on maximum Crit w/ both attacks: (5 + 6 + 6 + 6) + (5 + 6 + 6 + 6) = 46
Human Two Weapon Fighter with Dual Wielder feat (17.2 dmg/r, +6 to hit)
- +4 str bonus
- 1d8 longsword attack (8.5 dmg)
- 1d8 longsword attack (8.5 dmg)
- 5% crit (.45 dmg)
- Total damage on maximum Crit w/ both attacks: (4 + 8 + 8) + (4 + 8 + 8) = 40 Dmg
Best Answer
A creature can have Mass Polymorph cast on them to gain 157-247 temporary hit points
If your DM limits you to only beasts that come from core rule books (i.e. no Campaign books), then a targeted creature could take the form of a Giant Ape [Monster Manual, pg. 323], which would then give the creature 157 temporary hit points―but if your DM allows Campaign book beasts, they could instead take the form of a Sky Leviathan [Plane Shift: Kaladesh, pg. 28] which would give them a whopping 247 temporary hit points.
The only major restrictions are that creatures targeted must be at least CR7 (for the ape; CR10 for the leviathan), or the players targeted must be at least level 14 (for the ape; L20 for the leviathan). Additionally, the creatures will be required to sustain those forms in order to keep those Temporary Hit Points, so they won't be able to sustain those temporary hit points in their normal forms.
Honorable Mentions