A standard topic in a somewhat more advanced graduate course in complex analysis are elliptic functions. These are doubly-periodic meromorphic functions in the complex plane. According to Liouville's basic theorems, each elliptic function has to have poles and if there are only poles at the points of the period lattice, then they cannot be of first order. Now in his systematic theory of elliptic functions, Weierstrass introduced his zeta function as a meromorphic function with only first order poles at the points of a given rank-2 lattice. So this zeta function cannot be doubly-periodic according to Liouville. But can it be periodic? I will answer this question and show how this relates to many classical themes such as elliptic integrals, the hypergeometric ODE, Schwarz triangle functions, modular forms, etc. The talk will provide entertainment for a broad mathematical audience.