In the original Breath of Fire, Ryu's Dragon Forms fell into this category. The first 3 forms weren't so bad, but beyond that his dragon forms gave you the power to hit every enemy unit for more damage than most members of your party could hope to do to one, save perhaps with Bleu's high-end spells...which cost more to cast each turn than the one time you have to pay for Ryu's transformations. Oh, and if his HP hits 0? He transforms back to normal at full HP, ready to transform right back into that form good as new all over again.
This state of brokenness was so bad that in the second game of the series they went overkill on correcting the mistake and rendered them next to useless, becoming one-shot spells that sapped all of Ryu's available AP and, due to the damage they dealt being based on what percent of his maximum AP was used, made them even harder to put to good use as he leveled up.
If the Dragon transformations didn't interest you, though, Ryu also had another game breaker in the form of the Tri-Rang, the ultimate Boomerang-class weapon. It was one of his strongest weapons in the game, beat out only by the Infinity+1 Sword found in Jade's throne room by a pillar, and made up for this by hitting, you guessed it, every enemy on screen. What truly breaks it, though, is how stupidly easy it is to get once you're past the Guide Dang It: search the base of the tower on the northernmost island, just to the right of where you enter. No tricks, no traps, just DING, there you go. And if you're the type of player who reflexively pokes every nook and cranny for secrets and loot, it's not even Guide Dang It; it's just a matter of time before you find the correct spot, thinking something is probably there.
Breath Of Fire II
In Breath of Fire 2, we have the secret character Bleu. You can have her join your party as soon as you have access to the whale, and you get her at level 35 (for comparison, being around level 20 at this point is overleveled, albeit slightly). She has access to all of the best offensive spells in the game from the moment you get her, except for four of them, and she learns these four at levels 36, 37, 38, and 39 respectively. And her special ability heals her to full health at no MP cost - the only downside is a Defense debuff that wears off after the battle ends. And as she levels up, it gets beyond ridiculous. Other characters can typically expect to gain 7 or 8 points in their best stats, 1 or 2 in their worst, and roughly 4 on average. This character, when going from level 37 to 38, gains +15 HP +15 MP +15 Strength +15 Stamina +15 Agility +12 Wisdom +14 Luck. This is not an extreme example. This is normal for her. She gets +15 everything at least twice, and maybe it would be more if the stats weren't already capped by the point it could be possible. The only downsides are that she is a Guide Dang It to acquire and that she cannot bond with shamans (generally the way to power up your characters) - but really, she doesn't need to.
There's also the cook, who you can have access to as soon as you beat the Witch Tower (though he can be Lost Forever if you pick the wrong carpenter in Capitan). If you give him the right ingredients, he can cook extremely rare items that are massively gamebreaking when they're available in large quantities. The extremely rare item that lets you re-cast Ryu's uber-powerful Dragon summons which normally only work once per battle? Check. Items that boost your characters' stats? Check. Items that can be sold for more than the cost of their ingredients, essentially giving you infinite money? Check. An item that fully heals your whole party while simultaneously buffing them? Check. The only downsides are Guide Dang It and sheer tedium, but if you can cook enough of these items, you'll have a full party of uber-characters who when combined with shaman bonding can put the above-mentioned Bleu to shame.
Breath Of Fire III
It's rather easy to make a single character in Breath of Fire 3 into an invincible killing machine. Doing so is simply a matter of raising a character's Agility to a high enough level (either through leveling up under the faerie master Meryleep, who gives the highest Agility boost, or farming Swallow Eyes, which give a permanent + 1 to Agility). After that, stick the Resist skill (which makes the user completely immune to damage for the rest of the turn) on them. Thanks to BoF3's unique mechanics, any characters that are much faster than all the enemies get an extra turn. The "invincible" character simply Resists on their normal turn, and does all their attacking on their extra turns. You can also take further advantage of this by using the Chain Formation, which makes all three characters have the same Agility as the person in front.
Also from BoF3, Peco. Under the right conditions, he becomes just as impossible to kill as the Resist-user above. First, he has the highest HP and Defense in the game, and also average Attack. He passively recovers 5% of his max HP every round. He starts at level 1, which means you can get the maximum gains out of whatever master you choose for him. Most people use Fahl, who gives the highest HP and Defense gains at a level up, as well as some extra Attack. So several levels later, Peco is now pretty much Nigh Invulnerable. Now stick him at the front of the Attack formation, which both increases his attack power (moving it up from average to high) and makes enemies focus their attacks on him, letting our little Stone Wall soak up damage while the other two characters do as they wish without fear of getting killed. Oh, and don't forget that Peco ALSO has the highest reprisal rate in the game (a whopping 50% chance of countering physical attacks), which makes for great fun when he's getting hit with more than half the enemies' attacks? Watching your tiny, mutant onion slaughter the strongest enemies in the game is both awesome and hilarious to watch.
Don't forget - Aside from your main character being able to ascend into dragon form, which itself could be arguably a Game Breaker, Rei has the ability to transform into a Weretiger. Doing this greatly increases his physical power, but in return, you lose control of him and he has increasingly higher chances of attacking one of your allies each subsequent turn - Often resulting in a One-Hit Kill of that character. This is what normally balances it out. But add the Useless Useful Spell Influence which forces anyone who can be influenced to only attack whatever you targeted with the skill until either they or the target dies. This skill only affects a small number of weak monsters normally, yet it applies to Rei in this state, allowing you to designate a boss monster at the beginning of the fight and then leave him to continuously thrash it. Now combine that with the dragon form of your main character and commence the curb stomp of any enemy or boss, save for That OneBoss in Mook Clothing.
To screw things up even more, have Nina or Momo be your 3rd party member, then use the Fusion gene to transform Ryu into a tiger-dragon. On top of being faster than Rei, it also has access to the absurdly powerful Shadowwalk ability. Costs a scant 8 MP to use (making the form cost 16 AP per turn), and gives you a guaranteed critical hit. This, on top of Rei's spoileriffic abilities mentioned above, can even devastate the final boss with relative ease. Oh, and if you have the AP to spare, you can tack the Warrior gene on to ramp this hybrid form into Super Mode. At this point it will be more than half as powerful as True Kaiser form, and will get extra turns, making it insanely stronger and cheaper to use.
Once you get the Trance gene, Mook battles become nothing to worry about. For a piddly 13 AP to cast and 7 AP/turn to maintain, the Shadow/Trance combination grants Ryu access to the Tiamat form, which, on top of being nicely powerful and ridiculously quick, is a "convert the whole party" dragon form, adding in the benefit of the rest of your party being untouchable. It also has status-effect immunity. Oh, and a secondary breath weapon that hits every enemy with instant death and costs no AP to use. The awesome serpentine look is just icing on the cake.
The Tiamat form is also very effective against the final boss, being completely immune to her spammed poison/confusion spell, and beefy enough to survive until Ryu's AP runs out. It's not quite as flashy as transforming into Kaiser, but two transformations, a wisdom fruit, and normal attacks can win the battle with ease.
BOF3 also had a rare item called Ivory Dice that doubled experience from any enemy it was used on. With the aid of magical copying fairies, you could get lots of them, and stack them in battle to multiply any enemy's experience exponentially: 2 for 1, 4 for 2, 8 for 3, 16 for 4 and so forth. This is however balanced out by the fact that you can still only gain 65535 exp per battle and you need close to twice that to level up once you reach the high 60s, not to mention the extensive time investment in getting that many Ivory Dice to begin with.
Although it's only of any use in a TAS, you can get up to 32 hits with Super Combo, which at that point out-damages everything else in the game by a large margin.