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 players HP and MP levels. The new limit will be 9999 as that still remains the max HP limit.
Zack gains the Break MP Limit ability when equipped with the Genji Armor or with the Gold Hairpin. It extends the MP maximum to 9999. He can also break his max MP limit right from the start of the game with a strong enough 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 in the game.
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. This, in addition to the two magic casting-orientated characters having 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 International Zodiac Job System version, this can not be achieved since Quickenings use their own Mist gauge.