User blog comment:Edwin Shade/Defining LOCC (Language Of Ordinal Construction)/@comment-32697988-20170924234138/@comment-1605058-20170926213626

This only defines \(\omega\uparrow\uparrow n\) for finite \(n\). If you were to continue it by simply taking limits at limit ordinals, then, as Rpakr mentions, you would get \(\omega\uparrow\uparrow\alpha=\varepsilon_0\) for all \(\alpha\geq\varepsilon_0\) (which might or might not be what you want).