User blog comment:Ecl1psed276/Growth rate of oracle Turing machines/@comment-35470197-20190714085120/@comment-35470197-20190714231745

Remember that "a function bigger than \(f_n\) for any \(n \in \mathbb{N}\)" does not characterise \(f_{\omega}\) with respect to Wainer hierarchy, even if you consider the case where it can be obtained by "diagonalising" them. Say, FGH along \(\omega\) with respect to the funcdamental sequence \(\omega[n] = BB(n)+n\) also satisfies the condition, but is much bigger than \(f_{\omega}\) with respect to Wainer hierarchy. I guess tthat you understand that this FGH is bigger than \(BB\), and hence is not approximated by \(f_{\omega}\) with respect to Wainer hierarchy. Approximating \(f_{\omega_1^{\textrm{CK}}}\) with respect to Kleene's \(O\) by \(BB\) is quite similar to it.