TITLE: Function $f:\mathbb R^+\rightarrow \mathbb R^+$ that is eventually greater than $x^{x^{x^{...^{x^x}}}}$
QUESTION [8 upvotes]: 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.

REPLY [19 votes]: To make notation smoother, write $f(n,x)$ for $f_n(x)$. Let
$$f(x)=f(\lceil x\rceil, x).$$ 
Here $\lceil x\rceil$ is the "ceiling" function that gives the smallest integer $\ge x$. 
Remark: This is a typical "diagonalization" argument. Basically the same idea seems to have been first used by du Bois-Reymond to deal with orders of growth of functions. He did it a few years before Cantor used diagonalization in Set Theory.