User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-6768393-20130318025701/@comment-5529393-20130318123108

If Goucher's Xi function is indeed at the level of the first alpha such that alpha = omega_{alpha}^{CK}, then we can definitely exceed it - I describe how to get past it explicitly in my post.

Rayo's function is very, very large. It may be on the level of the largest countable ordinal definable in first order set theory, so the pitiful extensions that I made will be no match for it. However, it is quite likely that one can define ordinal notations and fundamental sequences, and therefore the fast-growing hierarchy, as far as one can define functions. Then the fast-growing hierarchy would extend past even Rayo's function.