These three spellcasting classes can prepare a number of spells equal to their relevant ability score modifier + their class level.
This means that at level 20 they could prepare 25 spells assuming they don't multiclass, their relevant ability scores are capped at 20 as per the class rules for Ability Score Improvements and they don't have any additional items or abilities that let them prepare more.
Is there any way to increase the number of spells a wizard, druid or cleric can prepare and if so, what is the maximum number that of spells they could prepared?
I know, for instance, that at level 20 a wizard gains the Signature Spell ability that allows them to always have two 3rd-level wizard spells prepared that don't count against the number of spells they can have prepared so that would effectively give them 27 spells prepared. Neither the cleric nor the druid have such an ability in the PHB and I haven't been able to find a magic item in the DMG that does this.
Are there any other, class/racial abilities, magic items, spells, etc. that allow for additional prepared spells? For the sake of this question I'm limiting it to official material only so no Unearthed Arcana or the like.
Best Answer
Magic Initiate
In addition to increasing your ability score beyond 20 (see ravery's answer), some classes can gain benefit from choosing the Magic Initiate feat (unfortunately, RAW, not those classes you specified in the question).
The feat states:
Per the Sage Advice Compendium, you can pick Magic Initiate for a class you already have and you can also use spell slots to cast the spell you gain more often, effectively giving you an additional 1st-level spell that you have always prepared.
Clearly, this allows classes that have all of their spells prepared (such as Sorcerers or Warlocks) to expand their daily available spell list.
Wizards are explicitly called out to require adding the spell to their spellbook, so (unless you houserule) they won't benefit beyond additional cantrips and getting a free casting once a day.
Druids or Clerics, however, don't have a spellbook and can already choose from all of their class spells when preparing. The rules are, therefore, unclear on whether they can count the spell gained from Magic Initiate as "always prepared". Judging by how Wizards have to prepare the spell, I believe an official rule would lean towards disallowing this.
Personally, I would houserule that the spell gained from Magic Initiate counts as an always-prepared spell, to avoid discrepancies between classes that have all spells prepared (e.g. Warlocks) and those that have to choose (e.g. Wizards or Druids). Especially Druids or Clerics would gain nothing, because - unlike Wizards - they wouldn't even benefit from an expanded spelllist.
You'd obviously have to talk about such a houserule with your DM, though.
Either way, Magic Initiate allows you to cast a 1st-level spell once a day for free, regardless of whether it's prepared or not. So, as long as you don't need the spell more than once a day (for example, choosing Alarm and casting it once a night), it's effectively an always-prepared spell.
Multiclassing
If you're either a Cleric or Druid, you can multiclass one level into the other class to gain a massive increase in prepared spells - in fact, you gain an additional number of prepared spells equal to your WIS modifier. Since the respective spell lists overlap in many cases, you can probably choose most of the 1st-level spells you want to prepare from the lower-level class's spell list, and choose the higher-level ones from your main class's list. Note that, technically speaking, you have to differentiate between both classes in terms of preparing, i.e. you can't prepare cleric spells with your druid "preparation slots". Therefore, you can only use the secondary class for low-level spells.
Unfortunately, this trick isn't as convenient if you're a wizard, since there is currently no other class that uses INT for spellcasting purposes (although this opportunity might come up if or once the Artificer or Mystic classes transfer from Unearthed Arcana to an official rulebook).
Multiclassing into a tertiary caster (such as Eldritch Knight or Arcane Trickster) is not viable, since they learn at most as many spells as their level - and every level you multiclass is substracted from the maximum number of Wizard spells you can prepare. In fact, multiclassing more than 4 levels into one of those classes will actually reduce your number of spells.
You can, of course, still multiclass into non-INT-casting classes to gain more prepared spells, but you'll then have two different spellcasting modifiers, which will mean the spells of the secondary class are weaker than those of your primary class.
The PHB states on page 164 on multiclassing:
Boons
Boons, as detailed in the DMG on page 232, can allow you to (sort of) increase your number of prepared spells. Namely, the Boon of Spell Mastery, Boon of Spell Recall, Boon of Dimensional Travel, Boon of the Fire Soul and Boon of the Stormborn allow you to cast certain spells for free.
Boon of Spell Mastery:
Boon of Spell Recall:
The other boons allow you to cast the spells Misty Step, Burning Hands and Thunderwave once per short rest or at will (depending on the boon).