Tetration towers

Consider the infinitely nested exponential equation

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

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

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

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

Give your answer to 3 decimal places.


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