Taking the powerset of the empty set N times, we end up with a set of size 2^2^...^2 (N times). A generating function would then take the form $$1+2x+4x^2+16x^3+65536x^4+ ...$$
I was trying to find a sufficiently differentiable function with such a Taylor series, but then I suspected that such rapidly growing Taylor series cannot possibly exist.
Can one please come up with such a function, or conversely, show that it doesn't exist?