Kingdom Hearts II has an area (namely, the Peak of the Pride Lands) in which many weak Heartless appear and respawn rapidly; with some experience boosters, Lv99 is a few hours away at most. The Final Mix removed this, however.
In Kingdom Hearts II, Berserk Charge is a very powerful skill on its own, but the drawback is that you never get a finishing blow on bosses. But when you combine this with a Square-button skill that acts as a finisher, such as Horizontal Slash...
Another good one for Kingdom Hearts II is the way MP works. With the right combination of abilities, you can basically make Sora near-invincible. To explain: Curaga heals Sora to full HP, at the cost of all his MP (and that's all his current MP, not maximum). This seems pretty balanced at first, but can become broken with the addition of two abilities: MP Rage and MP Haste. MP Rage makes your MP increase (or recharge faster) when you take damage. MP Haste makes your MP recharge faster when it runs out. These two abilities stack. Equip enough of them and you can basically get hit once and your MP shoots back to 100% again. Equip the Ultima Weapon (which is arguably much easier to get in the second game than in the first), which has the built-in unique ability MP Hastega, and add Leaf Bracer, which means Sora can't be hit out of casting Curaga, and as long as you can hit the shortcut fast enough you can basically never die.
Even better in Final Mix. By defeating one of the bonus bosses, you can get a synthesis recipe that for an accessory that boosts your strength, which isn't too bad. But upgrade it and you get the ability MP Haste. You can simply make 4 of these accessories and equip them all. That combined with the MP Haste you learn from leveling drive forms, MP Hastera you get if you play on critical mode, and the MP Hastega you get from equipping the Ultima Weapon, your MP will recharge absurdly fast. Useful for all those post game bosses.
The infamous "infinite combo" trick. Equip the ability Berserk Charge, which allows Sora to perform an infinite combo during MP recharge. Use Trinity Limit, a powerful attack that consumes all of Sora's MP, then follow it up with an infinite combo. Once your MP recharges, use Trinity Limit again and repeat the process until the enemy is dead. If you're fast enough, you can trap them in an infinite combo that they can't break out of. Sephiroth can withstand this thanks to his teleport powers, but even he can fall victim to this if you're fast enough and can lock back on to him immediately; it can even knock him out of his Heartless Angel attack!
Another one from KHII is Magnet/Magnera/Magnega. No enemy is immune (Sephiroth included). Non-bosses are sucked in, NO EXCEPTIONS. For added breakage, use it with Final Form.
Birth by Sleep has its own potentially game-breaking minigame. Basically, think Monopoly with four checkpoints you touch for points (don't have to touch them to win but they help a LOT). If you win a match, the abilities you used on the tiles that your enemies landed on will boost in the main-game and you may unlock more abilities. It is not quite game-breaking on its own but it can potentially make combat a bit too easy.
There's also a few spells in Kingdom Hearts: Birth by Sleep that could be considered Game Breakers, though none more so than the spell Mega Flare. It's an absolute nuke that hits the whole area for massive damage, and lasts for several seconds, meaning it's possible to damage multiple waves of spawning enemies with one spell. And, if you play the Command Board right, it can be obtained before you even get to the second world. Level up three Fires and two Aeros. Fuse the Fires into Firaga, then fuse the Aeros into Aerora. Now fuse Firaga and Aerora to make Fission Firaga. Make another Firaga (or buy one from a Moogle if you have enough money) and fuse it with Slow to get Crawling Fire. Now fuse Fission Firaga and Crawling Fire and proceed to own everything that stands in your way. Unfortunately, the spell was nerfed in the Final Mix version by doubling its recharge time.
Terra's combo finisher Meteor is another good one. A wide area of effect attack that hits for massive damage and makes Terra invincible during casting. Ventus and Aqua have Time Splicer, which freezes all enemies in the area and allows the user to warp between enemies while attacking them. Dodge Roll for Ventus and Cartwheel for Aqua are excellent for dodging, since not only do they allow you to move quickly out of harm's way, they also grant invincibility frames, meaning attacks can be survived by spamming Dodge Roll or Cartwheel. Also for Ventus, Salvation, which is basically his version of MM Miracle Lv3 from Kingdom Hearts: Chain of Memories — a massively powerful Holy attack that hits everywhere.
Even Salvation is dwarfed by Faith — a Ventus-exclusive command that deals heavy Holy damage and fully heals you. And unlike Salvation (which does more or less the same), it heals you fully, comes out immediately, and heals you at the beginning of the attack, rather than leaving you a sitting duck. The only drawback is that it is perhaps the hardest command to synthesize in the game — getting more than one is damn near impossible.
Perhaps the most simple of the gamebreakers in Birth By Sleep are the shotlock commands. Not only can do they do an utterly ridiculous amount of damage, they make you completely invincible for the duration of the move. With the Focus Guard ability equipped, you can max out your focus gauge in no time at all against tough bosses, and bombard them with shotlocks almost constantly.
Go on, set up a boss deck full of nothing but Sonic Blade + Multiple Hi-Potions in Kingdom Hearts:Chain of Memories. 90% of all boss battles can be won with this in under 2 minutes. Yes, all the way up to the 12th floor, at least. And there's only 13 floors. Needless, this is exploited like all hell for massive damage!
Even better if the boss has an element. Trouble with Vexen (who is ice elemental)? Throw a bunch of Firagas around + Multiple Mega-Ethers. Each Firaga takes off 3/4 of one of his HP bars. It's super-effective!
Actually, a Fire only deck is a game breaker in itself; a Fire only deck can make every boss but Axel (who is immune to it) a joke. Even the final boss in all his forms perish before it.
Omnislash (stack three Cloud cards in a sleight) for Sora and MM Miracle Lv3 (stack three The King cards in a sleight) for Riku. Omnislash is a powerful attack that hits everywhere and can only be blocked by a 0 card, which the AI can't seem to do as often as it would like. MM Miracle Lv3 is basically like a Holy version of the Phoenix summon from the Final Fantasy games - it does huge amounts of damage to everything in the area, heals Riku back to full HP, and fully reloads his cards. And since The King is Riku's only summon card, it's not too hard to get plenty of them. As for Sora, people have beaten the game by building decks containing nothing but Cloud cards.
Dark Mode is Riku's Game Breaker of choice. It powers him up considerably, while giving him access to three new moves: Dark Break (attacks from above), Dark Firaga (a powerful blast of energy that, despite its name, is actually dark-element, not fire-element) and Dark Aura, which is essentially the same attack he does when you fight him as a boss in the original Kingdom Hearts. All are achieved by stacking three Soul Eater cards in a sleight, the attack done corresponding to the total value of the three cards. The whole strategy for Riku basically involves getting him into Dark Mode as fast as possible.
In the PS2 remake, the safest and possibly quickest way to deal with many of the bosses is to simply abuse Riku's new Card Duel mechanic, which if successful, unleashes a very powerful attack that can't be canceled out by the opponent.
Also in the remake, Sora gets the new Lethal Frame sleight, which has a very simple combination (Stop + two keyblade cards), is very powerful, and most importantly, freezes the target a moment after its activation (even bypassing the immunity most bosses possess to the Stop status), which gives them only a very brief window of opportunity to cancel the sleight with a zero card; an opportunity that can be removed entirely if you break one of their cards with the sleight.
The original Kingdom Hearts can arguably be broken just by starting the game with "Shield Priority". With the ability to survive critical hits (that would have otherwise killed you) with 1 hp, the added HP, and perhaps some level grinding, even "Those Bosses" that are required to finish the game are no worry, due to the ability to tank the damage and just heal back up without a cooldown on the healing spell at all. Even "Magic Points" are not a limiter thanks to a skill that causes sufficient MP regeneration from damage taken. It is possible to rarely die, all the way to the end of the game, using this setup.
Starting with Shield Priority also allows Sora to learn the "Lucky Strike" ability much MUCH earlier than any other set-up, which means you can start collecting all the rare materials needed for forging better items (including the Ultima weapon) much earlier in the game.
The Balloon spell in Kingdom Hearts 3D. Available in the very first world, it's a low cooldown, high damage command that surrounds you with a cluster of stationary explosive balloons that detonate on contact with enemies, knocking them in the air and potentially stunning them. Multiple sets of balloons can be deployed simultaneously, and enemies and bosses are too stupid to avoid them, so it simply becomes a matter of waiting for them to go charging into their demise once you set them up. It's perfectly viable to have a command deck consisting of nothing but Balloon because of its sheer effectiveness.
And then there's its upgraded form, Balloonga, which launches a huge homing balloon that pops on contact, only to release six small homing balloons. It staggers most of the small bosses so easily that spamming it at them will keep mostly if not entirely unable to act, even the final bosses.
In Kingdom Hearts re: coded, there are the Triple Pursuit, Exo Spark, and Chain Rave commands. Triple Pursuit shoots out three homing ice crystals that pass through walls, and an enemy multiple times with the chance of freezing them, leaving them open For Massive Damage. Exo Spark surrounds you in a electrical field hitting multiple times. Chain Rave deflects incoming attacks, hits like a ton of rocks, and pushes you forward, allowing you to hit mid-range targets. All three are useful in filling the Overclock guage, and the only drawbacks for the former two are elemental resistances.
Judgment Triad, if by gamebreaking you mean "the Bonus Boss battle becomes dodge and shoot this thing". It has extreme homing that can potentially hit several Heartless at once, and damage so massive, it can one-shot several early-game bosses. You can only have 3 of these in your deck, but it was so good you barely need anything else.
Even stronger than all of the above is the Targeting Scope ability, which is the last clock ability on the Zero/One keyblade. It makes physical attacks automatically strike enemies at range. Add to the fact that earlier in the tree is an ability that makes every strike critical, and that Eliminator you've been running from is only so much road kill.
Kingdom Hearts: 358/2 Days's Mission Mode is completely broken once you're able to get Saix to a high level. His high strength and max crit chance destroys almost everything. His only downside is his low defence, but that shouldn't be a problem for someone who's beaten the game.
One for the story mode is the Ominous Gear, a gear that can be found in a chest in an extremely early mission, you just don't have the skills to obtain it at that point. It's MEANT to be gotten much later in the game, on day 158. However, by abusing a specific gear (The Lift Gear) and the Air Slide, it can be obtained on day 72 (74 if you've done all the side missions up to that point). Just for reference, the Ominous Gear is the Upgraded form of the Wild Gear, effectively the best Melee Based gear for the early game. You can then proceed to destroy almost anything for most of the game, until you get the local Infinity–1 Sword.