A bound state of electron and positron is well-known in nature and it is not stable as it will decay in two photons in 125 picoseconds for the singlet case. This is called positronium. So, photons should be regarded as truly elementary particles unless a more fundamental theory is found with other building blocks.
Anyway you can see this by yourself as electrons and positrons can have only weak and electromagnetic interactions and while the former cannot form bound states due to their extreme weakness, the latter forms bound states but with bound energy of the order of eV making the bonding not that strong. Besides, due to quantum electrodynamics, this bound is not even stable. You should note here a difference with respect to atoms.
Jon