User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-1605058-20130718061544/@comment-5529393-20130718171021

Of course, our notations are recursive, so we are limited to ordinals below the Church-Kleene ordinal. As we go higher and higher, though, we need more and more complicated hierarchies to collapse. For example, at the level of hyper-inaccessibles, we get a single collapsing function for each regular cardinal, whereas at the level of hyper-Mahlos we get a collapsing function for each ordinal in the notation. So we need our large cardinals to have sufficiently complicated orderings.

I think that the method of collapsing can go all the way to the Church-Kleene ordinal, though.