User blog comment:AJZajac/String Transforms (Outline - more detailed post to follow...)/@comment-25601061-20171016161117/@comment-27513631-20171020221841

Ordinal hyperoperators typically top out at \(\Gamma_0\) (the exception being Sibiis, who doesn't actually do ordinal hyperoperators), and in most formulations I've seen \(\alpha\uparrow^\beta\gamma \approx \varphi(\beta,\gamma)\) is frequent, so \(\varphi(\omega,0)\) is much more likely, assuming that the intuitive analysis is correct.