Tetration towers

Calculus Level 4

Consider the infinitely nested exponential equation

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

One might naively say, "Easy, just substitute in," and

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

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

Give your answer to 3 decimal places.