The photon is definitely massless. That is a theoretical prediction. Even a super strong one, actually.

There are experiments trying to falsify this statement. They have however so fair all failed, and the resulting upper limit on the photon mass is something like 10^-18 eV, which is about 1.7 x 10^-54 kg. The information can be found in the 2016 Particle Data Group review.

Thanks for your nice comment by the way!