We show that massive vector fields around rotating black holes can give rise to a strong superradiant instability which extracts angular momentum from the hole. The observation of supermassive spinning black holes imposes limits on this mechanism. We show that current supermassive black hole spin estimates provide the tightest upper limits on the mass of the photon ($m v<4 \times 10^{-20} eV$ according to our most conservative estimate), and that spin measurements for the largest known supermassive black holes could further lower this bound to $mv\lt 10^{-22} eV$. Our analysis relies on a novel framework to study perturbations of rotating Kerr black holes in the slow-rotation regime, that we developed up to second order in rotation, and that can be extended to other spacetime metrics and other theories.