User blog comment:Bubby3/FGH on ordinals/@comment-32876686-20171105152536

An interesting extension of FGH, but I think that $$f_2(\omega)=\omega\cdot 2^{\omega}$$, for the following reasons.

$$f_1{\omega)=\omega\cdot 2$$ $$f_1(f_1(\omega))=\omega\cdot 4$$ $$f_1(f_1(f_1(\omega)))=\omega\cdot 8$$

In general, the coefficient of omega increases by a factor of two each time. Hence, $$f_1(f_1(...f_1(\omega)...))$$, or $$f_2(\omega)$$ equals $$\omega\cdot 2^{\omega}$$

$$f_2(f_2(\omega))$$ would be equal to $$(\omega\cdot 2^{\omega})\cdot 2^{\omega\cdot 2^{\omega}}$$, and this would continue, so that $$f_3(\omega)=\epsilon_0$$, provided you define the fundamental sequence for $$\epsilon_0$$ according to the general pattern of $$f_2(\omega),f_2(f_2(\omega)),f_2(f_2(f_2(\omega))),...$$.