Of the 2 classes, fermions and bosons, only fermions can be their own antiparticles. Bosons are defined with having an integer spin, so they can never have spin 1/2. Of the fermions, none are known with neutral charge except for neutrinos, and we're not sure if those are Majorana particles or not.
Photons, as you mention, are bosons, with spin 1, so they can't be their own antiparticle.