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.

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:

Choose a class: bard, cleric, druid, sorcerer, warlock, or wizard. [...] In addition, choose one 1st-level spell from that same list. You learn that spell and can cast it at its lowest level. [...]

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.

If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat?

Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell.
If you have spell slots, can you use them to cast the 1stlevel spell you learn with the Magic Initiate feat?

Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.

In short, you must follow your character’s normal spellcasting rules, which determine whether you can expend spell slots on the 1st-level spell you learn from Magic Initiate.

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:

You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.

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:

Choose one 1st-level sorcerer, warlock, or wizard spell that you can cast. You can now cast that spell at its lowest level without expending a spell slot.

Boon of Spell Recall:

You can cast any spell you know or have prepared without expending a spell slot. Once you do so, you can't use this boon again until you finish a long rest.

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).

The Tome of Clear Thought will increase your Int and Max by 2 giving one more prepared spell to the Wizard.

This book contains memory and logic exercises, and its words are charged with magic. If you spend 48 hours over a period of 6 days or fewer studying the book's contents and practicing its guidelines, your Intelligence score increases by 2, as does your maximum for that score. The manual then loses its magic, but regains it in a century.

The Tome of Understading does the same for Wisdom giving the Cleric and Druid one more prepared spell.

From the Deck of Many Things:

Star: Increase one of your Ability Scores by 2. The score can exceed 20 but can't exceed 24

Wish can increase ability max but be careful:

State your wish to the DM as precisely as possible. The DM has great latitude in ruling what occurs in such an instance, the greater the wish, the greater the likelihood that something goes wrong.

While not technically Prepared Spells, Many magical staffs have a list of spells that can be cast from them.

Note: If you manage to gain multiple means of increasing your stat, 30 is the absolute maximum allowed.

Domain Spells, Circle Spells and Magic Initiate

Allow me to take some liberty and reinterpret this as "What is the most spells you can have prepared at 20th Level". This build will be mostly based on multi-classing to optimise the bonus spells provided by Cleric Domain and Druid Circle of the Land.

Optimise Ability Scores

Focus on getting 20 Wisdom and 20 Intelligence as quickly as possible. This build is purely about prepared spells so this isn't particularly hard to do by dumping physical abilities.

Nine Levels of Cleric

Each domain has a list of spells-its domain spells-that you gain at the cleric levels noted in the domain description. Once you gain a domain spell, you always have it prepared, and it doesn't count against the number of Spells you can prepare each day.

At level nine the Cleric has received all 10 of their Domain spells. The exact domain does not matter as they each recieve 10. Total prepared spells will be 9 (Cleric Level) + 5 (Wisdom) + 10 (Domain) = 24 Spells Prepared

Nine Levels of Land Druid

Once you gain access to a circle spell, you always have it prepared, and it doesn’t count against the number of Spells you can prepare each day.

As with the Cleric Land Druids receive the last of their circle spells at ninth level. However Druid's only receive 8 circle spells instead of 10 so the math would be 9 (Druid Level) + 5 (Wisdom) + 8 (Circle) = 22 Spells Prepared

Choice of Land Circle is not particularly important, however you should be careful not to overlap with spells provided from your Cleric levels.

Magic Initiate

Once your Wisdom and Intelligence are both at 20 you should take the Magic Initiate Feat.

Choose a class: bard, cleric, druid, sorcerer, warlock, or wizard. [...] In addition, choose one 1st-level spell from that same list. You learn that spell and can cast it at its lowest level. [...]

Magic Initiate Adds one addition spell you can have prepared. 1 Spell Prepared

One Level of Wizard

Unlike Cleric and Druid, Wizard does not provide bonus prepared spells so we only take 2 levels here. 1 (Level) + 5 (Intelligence) = 6 spells prepared

One Level of Any of the above

We have one level remaining and can take it in any of the prepared classes we have already taken. This will give us 1 additional prepared spell. 1 (Level) = 1 Spell Prepared

We could also choose to take a level in a different spellcasting class (Bard or Sorcerer) which would further expand our spell list. However in this context those spells would not count as "prepared" and won't help us achieve our goal.

54 Spells Prepared

24 (Cleric) + 22 (Druid) + 1 (Magic Initiate) + 6 (Wizard) + 1 (Other level) = 54 Spells Prepared

Once we have done all of these things we will have a massive 54 Spells Prepared each day. Unfortunately the highest level we can prepare is 5th for Cleric and Druid, and 1st for Wizard spells.

We would have all of the slot of a 20th level spellcaster however so plenty of slots to up-cast all those prepared spells.

This probably isn't a very useful build, and I'm sure I've missed something but this is the best way I know of to get the most spells prepared per day.

Increase Ability Scores beyond 20

Ravery's answer gives an excellent breakdown of the ways to increase ability scores beyond 20.

These magic items/spells would equally apply to this build. Particularly the Tome of Understanding which would gain us two spells (1 each from Cleric and Druid).

If by some miracle you managed to find/obtain all the items and spells he lists we would have a maximum of close to 60 spells!

Simulacrum

The 7th level Wizard only spell Simulacrum allows you to create a copy of a creature (including yourself), which has all the statistics (including spell slots, known and prepared spells) of the copied creature. Since there's no limit to the number of simulacrums you can have, you can just create as many as you need, swapping your prepared spells in-between the creation of each simulacrum (long rest needed for swapping).