User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130325203802/@comment-1605058-20130325205711

Nothing further than standard Kleene's O. It's trivial that this function maps recursive ordinals to recursive ordinals, so it doesn't take us further. On the other hand, we can define \(\mathcal{O}(7^n) := \omega_\mathcal{O}^{CK}\) this will take us pretty far into non-recursive ordinals