We derive and experimentally investigate a strong uncertainty relation valid for any n unitary operators, which implies the standard uncertainty relation and others as special cases, and which can be written in terms of geometric phases. It is saturated by every pure state of any n-dimensional quantum system, generates a tight overlap uncertainty relation for the transition probabilities of any n+1 pure states, and gives an upper bound for the out-of-time-order correlation function. We test these uncertainty relations experimentally for photonic polarization qubits, including the minimum uncertainty states of the overlap uncertainty relation, via interferometric measurements of generalized geometric phases.