Is there a closed form for the inverse of $y=x^{x^x}$?

Question

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any simple form of the inverse known?

Answer 1

Hint: Other super roots at Tetration wikipedia article.

Answer 2

The closest possible solution comes from noting $x^{x^x}=$ $^3x$.$$^3x=y$$$$x=\sqrt[3]{y}_s$$Also known as the super-cube root. Similar solution applies to higher towers.

Unlike $\sqrt{y}_s$, other super-roots are not definable in terms of known functions.