User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-5150073-20130811164029

Actually, the ordinal is higher than any recursive extension of \(\alpha \mapsto \omega^\alpha\) is $$\omega^{CK}_1$$, not \(\omega_1\) or \(\Omega\). The latter is the limit of any countable extension of \(\alpha \mapsto \omega^\alpha\).