Really, anyone who can do a barrier-breaking move can be considered as one. (Seriously, it's virtually required for the Final Boss, who has five barrier gauges!)
Take note however: Even with the following Game Breakers, this game is still at Nintendo Hard level thanks to the fact that bosses hit hard a lot, you don't get a lot of items to use, and your XP Gauge doesn't exactly match up to how they gain their XP. That and you're usually hitting up to 50k points of damage per unit at most when a lot of bosses are usually at 100k HP.
Chun-Li and Morrigan can be quite damaging in the right hands, and they learn skills faster than anyone else in the game with their first attack skill learned through level up being Chun-Li's Focus Attack that destroys barriers (if Chun-Li can't destroy it, then it's guaranteed that one of Morrigan's missiles will).
Ryu and Ken with the same input (down-A) can also enable them to destroy barriers as if they were paper.
Reiji and Xiaomu (yes, themagain) have Accel that guarantees they're going to be one of the first units to act and they learn the barrier busting skill that when you cast it, any first hit (they have to do the deed) is a Critical Hit, which will obliterate an enemy's barrier. Pair them up with Flynn who has a skill that gives +2 movement to all allies on the field and watch as hordes of mooks and bosses suddenly go down a lot more than usual and make pesky Timed Missions a lot easier to handle.
Their right-A attack can also break five-gauge shields - that's right, long before Chun-Li and Morrigan can get theirs.
Kogoro and Mii learn a skill for a lower casting cost that destroys an enemy barrier, the Tenzai Style: Ripple. What makes them a bit more special is that they can hit one more square away from the usual 2-square range. They also have an attack that destroys a five-gauged barrier that's available at the start of the game (Sharinken & Extensions), so it's more of choosing whether you want to break the barrier right away or do it the ol' fashioned way.
Heihachi's passive support skill allows a unit to laugh at enemies who have barriers since they can shatter them more easily, lessening the need to rely on the RNG.
Zephyr and Leanne's Multi Attack is nothing to scoff at; it is capable of inflicting a lot more damage, more powerful than what a typical Multi Attack should be. This is on top of them also having a skill that breaks through barriers like tinfoil. Then there's the fact that their ridiculously high hit count allows them to rack up XP faster than practically every other pair in the game, almost guaranteeing them at least one use of it every two turns.
Frank and Hsien-Ko\Lei-Lei work really well on any foe thanks to how simple the execution of their attacks are and their first attack skill that they learn being their Multi Attack. The only problem however is that they're a Mighty Glacier unit so they're almost always left behind. The realGame-Breaker is if you let them pair up with Alisa who has a skill that doubles the pair unit's speed, allowing them to at least go up near the first unit to act on a turn.
Akira and Pai have a skill call Stamping. Stamping can break barriers. That is all.
Haken and Kaguya have TWO attacks that can break barriers as soon as they are usable (neutral-A and left-A), combos that are easy to chain with their high TEC stat, the ability to attack from three squares away that can be increased to four if Haken's Sniping skill is used, and Love, the only skill which can heal all allies on the field to full health.
Project X Zone 2
Two words: Item Shop. Buy a ton of healing or SP recovery items (though you can only carry 15 per stage) and you won't have to worry too much about item management. Plus, the stuff sold at the shops is literally cheap and even if you fail or retire a mission you keep the gold you earn (sadly not the Exp or CP but that would be broken in of itself). It's so horribly broken that the game bans the usage of it in the challenge stages.
Stacking buffs is now a thing in this game. Want a character that can dish out so much damage? Go for it. Want a character to gain a lot of EXP? Check.
Stacking the charged attacks, using various attack buffs, getting behind a boss and make sure you have a super move usable. That way it's easy to do 70000+ dmg, even before you hit level 10: with enough buffs, you can kill any enemy in a single combo, even the final boss.
Axel's support skills is this, although a bit on the expensive side. He has a 150 SP skill that increases your attack by 100% (which you could get around just by equipping accessories to boost SP as SP is capped at 100 normally). While good, what makes him a Game-Breaker is that he has a skill that refills your entire cross gauge bar by 100% and it only costs 100 SP.
June's support skills are also powerful as her 150 SP skill increases attack damage by 25% to all units. Combine that with above, have someone be at full support, and you can easily break the game. Furthermore, nothing's preventing you from either spamming SP recovery items on whoever's paired with her or using the infinite SP trick described below, allowing you to cast the skill 8 times in a row, tripling your entire party's damage output until the end of the following turn.
For even more attack damage insanity, there's also Hibana's skill that costs only 100 SP, but boosts attack by 75% to her associated pair unit. Add in the potential to stack buffs, and you get a team that can cut down bosses and Elite Mooks like they were nothing.
Players can get an infinite amount of SP with the Reiji-Xiaomu, any other pair unit with Ulala, and Chrom-Lucina combo, similar to the Reiji-Aledycombo: Chrom and Lucina get a skill that increases anyone's max HP and SP by 50% and all other stats by 10% (A Game-Breaker skill in of itself for turning your teams into a Badass Army if you invest the time), Reiji and Xiaomu get a skill that fully refills anyone else's SP at a cost of 100 SP and Ulala starts with a skill that restores 50% of anyone else's SP at a cost of 50 SP. In short, Chrom uses his skill on whoever has Ulala, Ulala uses hers on Chrom to max out his SP, he uses his skill on Ulala again, at which point whoever has Ulala now has double the maximum SP, which Reiji can then restore, allowing both of them to provide infinite amounts of SP to anyone who needs it once Chrom uses his skill on Reiji as well. The only downside is that Reiji and Chrom will not be able to learn these sets of skills until they are at level 33 which usually won't happen until the last 10 chapters, three of those with all three separated, in the first playthrough unless you're using challenge stages to grind.
While Mirage Cancels are of marginal use at best during the first playthrough (you should just stick with the SP Attack for more damage and if your opponent is defeated you get more SP), they become by far the biggest Game-Breaker in the entire game in New Game+ where you can use them as many times as you want in a single combo: you can use your carryover CP to instantly max out any move that recovers a lot of XP when used, stack the XP recovery buff to triple your XP gain and use any combination of equipment and skills to give you -30% Mirage Cancel cost and you can essentially take down any enemy in a single combo regardless of how much HP they have as long as you can time the Red Mirage Cancel correctly. Zephyr and Vashyron benefit from this the most because of how many hits they can land in their combos.
Overlaps with Bragging Rights Reward: each Pair Unit has two pieces of exclusive equipment which can be obtained by completing each of the challenge maps; the last one has the strongest dropswhich can be used by any team. Granted, because of the higher levels of the challenges compared to the main campaign, you'd need to make use of some of the actual techniques listed here to survive, such as the Reiji/Xiaomu, Ulala and Chrom/Lucina strategy and Axel's 150% attack increase. Depending on your strategies, you can use these maps to get the best equipment possible and grind for experience; combining those two together will absolutely destroy the story.
The Challenge Maps themselves are this for another reason, it is the fastest way to earn gold. Besides the Level Grinding and the equipment mentioned above, you can repeat the challenge maps to sell the extra equipment from 100,000 to 300,000 gold depending on the level and in the last challenge map, the above strongest equipment is worth 500,000 each piece. This can really add up and soon you will get to the point of having Money for Nothing.