User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-24509095-20140507175906/@comment-24509095-20140508112129

Nope. It doesn't always apply: $$\psi_1(\psi_1(\psi_1(\psi_1(...))))=\psi_1(\Omega_2)$$.