User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-5150073-20130414231904/@comment-1605058-20130417124555

Not every ordinal can be defined using hyper-operators. For example, \(\phi_(\omega,0)\) would require \(\omega\) up arrows, so is beyond standard hyper-operators. Of course, we can extend this, but there must be ordinal \(\Gamma\) such that \(\Gamma=\omega\^{}\^{}^{...}\omega\) with \(\Gamma\) arrows.