User blog comment:Edwin Shade/Hyper Operations On Ordinals Using The M.O.S.T. Function/@comment-11227630-20180428062354

I got $$2\uparrow^\omega\omega=\sup\{2\uparrow^n\omega|n<\omega\}=\omega$$, where $$2\uparrow^n\omega=\sup\{2\uparrow^nm|m<\omega\}=\omega$$.

Also, $$\omega\uparrow^\omega2=\varphi(\omega,0)=\omega\uparrow^\omega\omega$$.