David Z answers a part of your question great, so let me fill in the other part:
does that mean there is a theoretically maximum brightness in which no
additional amount of photons could be present in the same volume
(Disclaimer: I'm just an amateur - this is the way I understand the subject matter)
The answer is mostly a yes. While as David says, photons are bosons and therefore do not really have a meaning of "personal space", the accumulation of the photon's energy causes another very interesting thing to happen - the spontaneous creation of new particles. In fact, there's a kind of supernova that's theorized to occur because of this - http://en.wikipedia.org/wiki/Pair-instability_supernova.
In theory, this can happen as soon as there's enough energy to produce any pair of particle-antiparticle. However, at the same time, there's considerable support for the idea that for a short time after the big bang, everything was photons - that was probably the biggest amount of photons in the smallest space ever. At that point, the universe was too "hot" to enable pair production.
So under "normal" conditions, the amount of photons in a given volume is constrained. However, it's not about the amount of photons - it's about their total energy.