Tetration towers

Calculus Level 4

Consider the infinitely nested exponential equation:

$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} = N$

One might naively say... Easy, just substitute in, and:

$$x^N = N$$, so $$x = \sqrt[N]{N}$$.

However, this doesn't converge for all $$N$$. What is the highest $$N$$ for which it does?