What is a gentei kaijo?

Answer 1

It is the supreme level of power for a soul reaper. This sort of power has to be authorised by the soul society council before any soul reaper can tap into it. With this new, rejuvenated power, they can increase their speed, combat, zanpakto, zanjistsu etc by 10 fold.

It was first introduced in the Bleach Anime series 6 episode 120 when they were fighting against new foes the Arrancars

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Edit (former answer not entirely accurate): Gentei Kaijo in English means "limitation cancellation". Found in the anime "Bleach" when lieutenants and above go to the world of the living, their powers are limited to 10%. A seal is placed on their bodies, and once the soul society has placed proper protection around nearby souls, they will approve the release of this seal, allowing the soul reapers to fight at their natural full strength. So it's not a new level, or additional powers, it means breaking down the dam holding back their own true power.