User blog comment:Nayuta Ito/The attempt to make my own ordinals collapse function/@comment-28606698-20170422072446

I think growth will be slightly faster, namely:

$$\rho(3)=\omega^{\omega^2}$$,

$$ \rho(n)=\omega^{\omega^{n-1}}$$,

$$\rho(\omega)=\omega^{\omega^\omega}$$,

$$\rho(\omega+1)=\omega^{\omega^{\omega+1}}$$,

$$\rho(\omega\uparrow\uparrow k)=\omega\uparrow\uparrow (k+2)$$.

If you will exclude multiplication, leaving only addition, it would be

$$\rho(\omega\uparrow\uparrow k)=\omega\uparrow\uparrow (k+1)$$.

and thus we can say that functions with and without multiplication catch one another near $$\varepsilon_0$$