Does there exist an increasing function $f$ $>0$ for all $x> 0$ such that $$f(2x)=2^{f(x)} \text{ for } x>0?$$
I first considered sequences of the form $u_0=a$, for $a>0$ and $u_n=2^{u_{n+1}}$ to find $f(\epsilon)$ for $\epsilon$ very small but it doesn't work (not an infinite sequence).
No, there is no such function. Given your assumptions $f$ could be prolonged in $0$ by continuity (as $f$ is positive and increasing). But $a=2^a$ has no solution for $a>0$.