For each $n$, define $f_n:\mathbb R^+\rightarrow \mathbb R^+$ by $f_n(x) = \underbrace{x^{x^{x^{...^{x^x}}}}}_n$
I want to find a function $f:\mathbb R^+\rightarrow \mathbb R^+$ such that for any given $n$, $f$ is eventually greater than $f_n$.
Here $\mathbb R^+$ means the non-negative reals.