User blog comment:LittlePeng9/Functions between computability and busy beaver/@comment-1605058-20141130153958

I wonder if something like this is possible for some analogue of this for levels of Hardy hierarchy, so for example, a function \(F(n)\) dominating all of \(H_{\omega\uparrow\uparrow k}(n)\) but no recursion below \(\varepsilon_0\)  of \(F\) is enough to outgrow \(H_{\varepsilon_0}(n)\).