Do photons occupy space?

However, they must somehow occupy space, as I've read that light waves can collide with one another.

That's not true. Yes, light waves can "collide" and interact with each other (rarely), but that itself doesn't imply that they need to occupy space.

It's not even entirely clear what it means for a subatomic particle to occupy space. A particle like a photon is a disturbance in a quantum field, and is "spread out" across space in a sense; it doesn't have a definite size in the same sense that a macroscopic material object does. But you'll probably agree that, if it's possible to make any sensible definition of "occupying space" for a subatomic particle, it should involve preventing other things from also occupying that same space. Photons don't do that. They're bosons, and as a consequence of that they are not subject to the Pauli exclusion principle, so if you have a photon occupying some space (whatever that may mean), you can in theory pack an unlimited number of additional photons into the same space.

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.