User blog comment:PsiCubed2/For Newbies (and Veterans too): The Great Scale of Googology/@comment-25601061-20171219161544/@comment-1605058-20171221122634

Would you mind linking the article claiming oracle TMs reach \(\omega_2^\mathrm{CK}\)? There might be some mistake or a misunderstanding there. For the sake of reference, let me mention this Other Wiki article, in particular the last paragraph of the linked section - it implies thay every \(\Sigma_1^1\)-definable notation is bounded below the CK ordinal, and things recursive in the halting oracle are \(\Sigma_2^0\) (never mind what those mean exactly :P)

As for MathJax, I've noticed that it always loads fine if you enter the blog post via a link to a comment; otherwise, the comments section might load after MathJax is done loading.

I'm afraid the next question is ill-posed - I have no idea how to define the "naive computable extensions" of the CK ordinal, which would somehow amount to extending an arbitrary recursive function to the CK ordinal, for which there is no general method. Overall, I'm afraid the "landscape" between \(\omega_1^\mathrm{CK}\) and \(\omega_2^\mathrm{CK}\) has not been really explored by anyone.

To the last question: you can take the BB functions for Turing machines with the hyperjump oracle, for example. Noteworthily, the countable admissible ordinals are precisely the ones which are limits for TMs with some fixed oracle.