Quantum mechanics implies properties of Nature that clash with our intuitive notions of how the universe ought to work. Testing these properties (to see if quantum mechanics is, indeed, true) involves generating entangled quantum states of two or more particles and measuring them under a number of strict conditions. While work is progressing to meet all of these conditions when using only two particles, no one has yet met even one of these conditions for more than two particles, which is considerably more difficult experimentally. Here, we conduct an experiment where we meet two of the most challenging conditions—namely measurement locality and freedom of choice—while generating triplet entangled photon states. We demonstrate that quantum mechanics wins out over intuition, measuring a violation of Mermin's inequality outside the classical bound by nine standard deviations.