electromagnetic radiation - Do photons have a maximum density per unit area?

I've been sitting on this thought for a while, and I figure it's time to pass it off to someone more educated than myself.

Consider the following hypothetical scenario:

In a vacuum, we have six panels, each of which has one face which is a perfect white body (known to the layman as "a mirror"). The other face is translucent - it allows some light through, though the exact amount is irrelevant. These panels are similar to the two-way mirrors seen in police interrogation rooms on TV when viewed from the observer's room - essentially, light can enter the observer's room, but it can't leave. Any light that hits the panel that would otherwise enter the interrogation room is reflected back "completely and uniformly in all directions" (per wikipedia definition of a white body).

Using these six panels, we create a box where all white body faces are directed inwards, such that if we shine a light at the box from the outside, light can enter the box, but it cannot leave the box.

If we shine the light at the white body box for a short amount of time, and then turn it off, the photons (should) bounce around inside the box indefinitely.

If we shine the light at the white body box and do not turn it off...?

Question: Does light have a maximum density per unit area? (Can we "fill" the box with light? If so, what happens? Why?)

Bonus Question: Does the effect differ if we were to shine a different wavelength of light (e.g. radio wave, xray, etc.)?

PS: I'm aware that Density may not be the proper term to describe the question I'm asking. The idea of whether or not light has mass seems ambiguous to me - while it seems to be the general consensus that light does not have mass, it does have energy, so... let me know in the comments.

PSS: I'm aware that my box must be constructed of unobtanium (read: doesn't exist), but just bear with me.

Answer

Every photon has the energy $e=h\cdot f$ where $h$ is Planck's constant and $f$ the frequency. Since mass and energy are related through $m=e/c^2$ the mass of your box would become larger the more photons you put in. Since the Schwarzschildradius is $r_s=2\cdot G\cdot m/c^2$ the Schwarzschildradius of the box would after a while exceed it's radius and become a black hole, assuming of course that no radiation leaks out and your box can handle the temperature and pressure. If for example you heat up a can of water with a laser it also gains mass, which it radiates away when it cools down. If you shine in more than your box can radiate away it will become heavier. Since photons are bosons which can in contrast to fermions co-exist in the same state in the same place and time there is in priciple no upper limit of how many photons you can put into your box.