what is the radius of convergence of the sum (n=0 to infinity) of z^(n!) ?

Certainly it converges for all |z|<1 since it is a subseries of the series z^n and certainly it diverge when z=1.

 

But a more interesting exercise: show it diverges for a dense set of points on the circle |z|=1, thus it can how no continuation beyond the unit circle. This is what makes it an intersting function.