User blog comment:MachineGunSuper/Question/@comment-30118230-20180224173316

BEAF is not fully well-defined. It is only defined up to tetrational arrays, so it's limited by $$\varepsilon_0$$ in the FGH.

The FGH can be possibly extended infinitely so far as there are ordinals for it and proper fundamental sequences for them. The FGH can even be uncomputabely strong if there are ordinals beyond $$\omega^\text{CK}_1$$ with proper fundamental sequences.