User blog comment:MachineGunSuper/Question/@comment-32783837-20180224123408

FGH can be extended potentially infinitely, as long as there are notations capable of creating larger and larger computable ordinals. (*cough* Taranovsky's C)

BEAF is ill-defined after X↑↑X & n, and so it is limited at the equivalent of ε0 in FGH.