I've been trying to solve the equation:
$$x^{x^6}=\sqrt{2}$$
by Newton's method, I got the answer to be somewhere around 1.1562. I was wondering if there was a way to solve it analytically, you know getting an exact answer. I've been trying for a couple of hours now, but any exact solution eludes me.
As said in comments, there is an analytical solution in terms of Lambert function $$x^{x^a}=b \implies x=\sqrt[a] {\frac{a \log (b)}{W(a \log (b))}}$$