Is generating function defined for every possible sequence?

Question

For every converging series surely it is, but how about $a_n=2^{2^n}$? Is $f(x)=\sum_{n>=0}2^{2^n}x^n$ well defined? It actually converges only for x=0, but my intuition says it is somehow not well defined. If it's perfectly fine to make generating function of $a_n=2^{2^n}$ is there some other weird seqence that generating function is not defined for?