In contrast to other topics in analysis such as functional analysis with its vast amount of counterexamples to intuitively correct looking statements (see here for an example), everything in complex analysis seems to be very well-behaved (for example holomorphic functions are always analytic). But is this maxim always right? Do you know any holomorphic functions which behave in a way one wouldn't expect at first sight?
EDIT: As you can see in the answers, I came up with something myself. But I would be glad if you knew more examples where strange stuff happens.