User blog comment:Bubby3/Ridiculously strong ordinal hyper-operations/@comment-28606698-20170612221214

Since you are referring for Saibian I just note that he wrote

$$\omega\uparrow\uparrow\omega^2 \rightarrow \varphi(2,0)$$

$$\omega\uparrow\uparrow\omega^\omega \rightarrow \varphi(\omega,0)$$

$$\omega\uparrow\uparrow\uparrow\omega \rightarrow \varphi(1,0,0) = \Gamma_0$$

$$\omega\uparrow\uparrow\uparrow...\uparrow\uparrow\uparrow\omega\rightarrow\varphi(\omega,0,0)$$

What is in your opinion $$\omega\uparrow^{\alpha+1}(\beta+1)$$, for example $$\omega\uparrow\uparrow(\omega+1)$$?

for $$b<\omega$$ we can define $$\omega\uparrow\uparrow(b+1)=\omega\uparrow(\omega\uparrow\uparrow b)$$

but if $$b=\omega$$ then according this definition we have $$\omega\uparrow\uparrow(\omega+1)=\omega\uparrow(\omega\uparrow\uparrow \omega)=\omega\uparrow(\varepsilon_0)=\varepsilon_0=\omega\uparrow\uparrow\omega$$