Any character equipped with the Independent Materia HP↔MP obtained at the Materia Cave closest to Mt. Corel, may break the MP limit if they possess a Max HP of over 999, as it reverses the player's HP and MP stats. The new limit will be 9999 as that remains the max HP limit.
Zack gains the Break MP Limit ability when equipped with the Gold Hairpin or with the Divine Slayer. It extends the MP maximum to 9999. He can also break his max MP limit right from the start with a strong enough Digital Mind Wave Limit Break, or power surges from said Limit Break, albeit capping at 999 without the Break MP Limit.
Break MP Limit is the third-most dominant ability, and fourth-most dominant ability combination. While armor equipped with Ribbon will be named based on the Ribbon auto-ability, when paired with the second-most dominant ability, Break HP Limit, it becomes the most dominant ability combination.
The Shield for Tidus with Break MP Limit is the Emblem. The Ring for Yuna is the Mythical Ring. The Bracer for Auron is the Dragon Lord. The Armguard for Wakka is the High Spirits. The Armlet for Kimahri is the Sage's Armlet. The Bangle for Lulu is the Mythical Bangle. The Targe for Rikku is the Celestial.
When paired with Break HP Limit, Tidus has the Endless Road, Yuna has the Solomon Ring, Auron has the Overlord, Wakka has the Indomitable, Kimahri has the Pride of the Ronso, Lulu has the Samantha Soul, and Rikku has the Dreadnought.
Break MP Limit isn't considered a very useful ability, since MP isn't consumed that quickly, Quick Hit being the most commonly used MP attack in the later stages of the game. Also, the two magic casting-orientated characters have the One MP Cost ability equipped to their respective Celestial Weapons.
Characters can break their MP limit by obtaining second and third Quickenings, doubling and tripling the character's maximum MP respectfully. The character's max MP will still read as the orthodox number, however, their MP gauge will be based on the actual MP number, and the player can still restore MP up to the actual MP number.
In the Zodiac versions, this cannot be done since Quickenings use their own Mist gauge. The amount of MP gained upon leveling up is increased significantly to compensate, and it is possible to reach 999 MP, but a hard cap is set.