User blog comment:PsiCubed2/For Newbies (and Veterans too): The Great Scale of Googology/@comment-25601061-20171219161544/@comment-1605058-20171221133738

I would disagree about that statement itself being confusing - there is nothing in that claim which indicates anything about the halting oracles in particular. What I believe got you confused is the impression (which people around here seem to share and which I'm not sure where it originates) that "oracle Turing machine" means by default Turing machines with the (possobly iterated) halting oracle.

Either way, I do have to agree that googologists' needs are not going to be satisfied with this account of admissible ordinals, leaving basically nothing inbetween. What you need is, as in the recursive case, a finer scale of functions, and unfortunately computability theory as generally studied is not going to be an ally of yours in this.