Is a function still analytic if its infinite sequence of subsequent derivatives, at every point of its domain, grow faster than the factorial?

Question

Taylor series have the nth derivative ($f^{(n)}(a)$) in their numerator and n! in their denominator. I was wondering what if the derivatives (for any point $a$) grows faster than the factorial. Does the function still equal it's Taylor series? Also, give examples of such functions.