User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-1605058-20171029165853

If we just artifically add our function \(f\) at the top of the hierarchy, then there is little reason to expect that it fits the limit rule of FGH. It's just like trying to add some arbitrary function in place of \(f_\omega\); it just doesn't "fit".

Regardless, it is possible to have \(f_{\omega_1^\mathrm{CK}}\) computable, it is even possible to have it equal \(f_\omega\), by a modification of an argument given here.