Indistinguishability between photons is a key requirement for scalable photonic quantum technologies. We experimentally demonstrate that partly distinguishable single photons can be purified to reach near-unity indistinguishability by the process of quantum interference with ancillary photons followed by heralded detection of a subset of them. We report on the indistinguishability of the purified photons by interfering two purified photons and show improvements in the photon indistinguishability of 2.774(3)% in the low-noise regime, and as high as 10.2(5)% in the high-noise regime.