User blog comment:PsiCubed2/For Newbies (and Veterans too): The Great Scale of Googology/@comment-25601061-20171219161544/@comment-1605058-20171221112411

I don't know how strongly this depends on fundamental sequences (and I don't want to start this discussion again now), but it's a matter of fact that Turing machines with a halting oracle don't reach \(\omega_2^\mathrm{CK}\) in any reasonable way - the well-orderings computable by a TM with a halting oracle must have length smaller than \(\omega_1^\mathrm{CK}\). The same holds for halting oracles iterated any finite or recursive number of ways. With this in mind, I don't know of any sensible way to assign "ordinal growth rates" to Busy Beaver oracle functions.

For SKIO calculus, I don't know whether it is possible to compute any longer well-orders. However, here I outline the proof that SKIOO2 (O2 is an "is well-founded" oracle) can compute well-orderings of length up to \(\omega_2^\mathrm{CK}\), and more generally SKIOO2...On should reach at least \(\omega_n^\mathrm{CK}\). No idea how to reach anything even remotely like \(\varepsilon_0^\mathrm{CK}\), let alone with SKIO itself.