Let $$r(f, x) = k$$ such that

$$f^k (x) < 2 \hbox{ and } f^{k-1} (x) \geq 2$$

For example $$r(n \rightarrow n-1, 2^n) = 2^n-1; r(n \rightarrow n/2, 2^n) = n.$$

For an arbitrary $c$, we have $$r( n \rightarrow n/2^{1/c}, 2^n) = cn$$.

Question, for an arbitrary $c$, is there some closed form function $f$ such that

$$r(f, 2^n) = n^c$$ ?

Thanks!