Can you define a elementary real-valued function $f$ such that $2^{x} < f(f(f(x))) < 2^{2^x}$ for sufficiently large $x\in \mathbb{R}$?
I know that there is no elementary function $f$ such that. $f(f(x))=2^{x}$ but is it possible to find an elementary function such that $2^{x}<f(f(f(x)))<2^{2^x}$ for sufficiently large $x\in \mathbb{R}$?