R = N ^ (N ^ (N ^ ...)). What is the maximum N>0 that will yield a finite R?

Since R=N^R, we need to find the largest N such that the curves f(R)=R
and g(R)=N^R have a point of intersection for some N>0. For
small N, these curves intersect at two points, while for large N, they
don't intersect at all. The maximum N is such that they intersect
at exactly one point.

At this point, the curves are tangent; since f has slope 1, we have
g'(R)=1, or ln(N)*N^R = 1. Plugging in R=N^R gives
R*ln(N)=1. But we can also take the ln of both sides to get
ln(ln(N))+R*ln(N)=0, so that ln(ln(N))=-1. The maximal N is thus

e^(1/e).