Results of a search for the pair production of photon-jets - collimated groupings of photons - in the ATLAS detector at the Large Hadron Collider are reported. Highly collimated photon-jets can arise from the decay of new, highly boosted particles that can decay to multiple photons collimated enough to be identified in the electromagnetic calorimeter as a single, photonlike energy cluster. Data from proton-proton collisions at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 36.7 fb-1, were collected in 2015 and 2016. Candidate photon-jet pair production events are selected from those containing two reconstructed photons using a set of identification criteria much less stringent than that typically used for th...