Is there an exact term for $\sqrt{2+\sqrt{4+\sqrt{8+\dots}}}$

I'm wondering whether it's possible to find an exact term for the infinite nested radical expression from the title. I got a quite good approximation with my calculator but what I'm looking for is an exact term.

$$ f(x)=\sqrt{2^x+\sqrt{2^{x+1}+\sqrt{2^{x+2}...}}} $$

It is necessary that f satisfies the condition: $$ f(x)^2=2^x+f(x+1) $$

EDIT: But there should be infinetely many solutions to this equations - furthermore, I wasn't able to find a single one!

Does anyone have an idea how to find an exact finite term - or prove that no such term exists?

Solution 1:

Related will be $$ g(u) = \sqrt{u+\sqrt{2u+\sqrt{4u+\sqrt{8u+\dots}}}} $$ so that $f(x) = g(2^x)$. Graphically, $g$ looks like this,
with $g(2) = 2.17968$ marked. Numerically we see $$ \lim_{u \to 0^+} g(u) = 1 $$ $g(u)$ is complex (non-real) for $u<0$.

$g$ satisfies $g(u)^2=u+g(2u)$, which lets us compute an asymptotic expansion for $u \to \infty$ $$ g(u) = u^{1/2} + \frac{1}{\sqrt{2}} +\left(\frac{1}{2\sqrt{2}}-\frac{1}{4}\right)u^{-1/2} +\left(\frac{1}{8\sqrt{2}}-\frac{1}{8}\right)u^{-1} +\left(\frac{9}{32\sqrt{2}}-\frac{3}{16}\right)u^{-3/2} +O(u^{-2}) $$