This Wikipedia article (on Morera's theorem) mentions the following in the introduction.
In a certain sense, the $\frac{1}{z}$ counterexample is universal: for every analytic function that has no antiderivative on its domain, the reason for this is that $\frac{1}{z}$ itself does not have an antiderivative on $\mathbb{C}\setminus\{0\}$.
Why is this statement true? In particular, what is the connection between the failure of an arbitrary holomorphic function to have a primitive and the failure of $z\mapsto\frac{1}{z}$ to have a primitive? Perhaps this is related to term-wise integration of Laurent series?