Every function I've come up with, which has a branch point, e.g. $x^\beta(\ln x)^\gamma$ at $x=0$, after several repeated differentiations appear to have an infinite limit at the branch point — at least from some side. I guess it's a common property of branch points that eventually their derivatives become infinite, given a high enough order of the derivative.
Is this correct in general? Or are there any other types of branch points, derivatives of which always remain finite?