This has already ben beaten into the dirt, but I’ll just add that the number of reflections you’ll get with mirrors having a reflectivity of R is about R/(1 - R), so if you have Chronos’ 99.99% reflective mirrors ( R = 0.9999), you’d get about 10,000 bounces as a characteristic number. If your mirrors were a meter apart, with light going at 3 x 10[sup]8[/sup] meters/ second, that would last about 33 microseconds, which a pretty impressively long time by the standards of optical things, but still incredibly short by human standards.
Surface imperfections, dirt on the surface, any sorts of minute imperfections will contribute to your scattering. Your ideal reflectors sound like Larry Niven’s stasis field from his Known Universe stories. As he claimed, it would be a perfect reflector, and could be made ideally flat and scatter-free. But I’m not sure you’d find it anywhere else.
What if you had perfectly flat, reflective, parallel and infinitely large mirrors? 
Actually, as I reflect on it (Ha!), there’s an interesting fillip to this. At each bounce, the photon will be transferring momentum to the mirror. If it’s perfectly reflective, it will rebound with exactly the opposite momentum that it started with, going from p to ** - p**. If it rattles back and forth, it keeps doing this forever, which seems uncomfortably close to perpetual motion, or something.
There’;s a discussion of this here, considering the case for solar sails. Obviously it has bothered other people, too:
They are correct, I think, in stating that a photon bouncing off the solar sail will undergo a tiny shift in frequency. You gotta pay for the momentum transfer somehow. But what does that mean for our photon between fixed mirrors? To the photon, the mirror looks just like a solar sail.
It seems to me that what happens is this – you’ve got your two mirrors rigidly attached to each other. Let’s assume that the mirror box is floating in free space. The photon bounces off one mirror, it transfers p momentum to the mirror construction and undergoes a tiny frequency shift as it rebounds with ** - p** and goes back the other way. Now it strikes the other mirror. The other mirror, however, isn’t stationary – you transferred momentum to it when the photon bounced off the first mirror. So now the photon bounces off a moving mirror. It gains energy, being frequency shifted back to close to where it was initially. The mirror box loses most of its momentum its momentum. So you end up with the photon gaining and losing minuscule amounts of frequency with each bounce, but returning to the same value. The box ultimately isn’t going to move, because its a contained system with no external forces, so the center of mass is going to stay where it is. Photons have momentum, but not mass, so the box has to stay put. Its apparent motion (since it is successively gaining momentum and losing it) is probably a subtle relativistic effect that ends up as a washout. In the real world, the momentum imparted to a mirror you can easily pick up and handle is down in the thermal noise.
I think the use answer to the question is to start with just the idea of bouncing a photon about, and then to enumerate all the ways in which is won’t stay bouncing around, and, in the quest to make the system “perfect” you can start positing a miraculous system where each mechanism in turn is eliminated, and simply look at the relative proportions of the mechanisms left.
So you have (a non exhaustive list):
[ul]
[li]geometric imperfection causing loss of the photon from the system[/li][li]reflective imperfection/contamination causing adsorption[/li][li]thermal and other random events that cause loss of geometric perfection[/li][li]All the QED possible paths that cause the photon to be lost.[/li][/ul]
Some of the QED paths can be thought of as tunnelling, some will involve an electron managing to reach an energy state that is able to do something unusual with the photon (I think). You can sit down, start drawing Feynman diagrams, work out the path integrals, and thus derive the probabilities if you want. A real physicist could make a much more useful summary here.
Probably the nearest physical system that touches on the OP’s ideal is an optical fibre. Not quite reflecting, but it does trap a photon for quite a while.
If the mirrors are attached to each other, then the box would wiggle back and forth, with no bulk motion: When the photon is moving to the left, the structure of the box is moving to the right, to cancel out its momentum, and likewise when the photon is moving to the right, the structure of the box is moving to the left.
At least, on average. You’re not going to get the whole box wiggling in unison, because the transfer of wiggle from one mirror to the other is going to be through sound waves through the struts connecting the mirrors. Sound is slower than light, so by the time that the left mirror finds out that the photon hit the right mirror, the photon has already bounced off the left mirror and is on its way back right (probably a great many times).
How long would it take to do this?
At 93,000,000 miles apart it would give you about 8 minutes.
Where was he going? I assume it would be some sort of 'ahhha we can make infinite magical energy" or somesuch.
I am sure the best way to answer the OP would be 'what would you like the outcome to be" and go from there just to get past the inevitable string of ‘well let’s ignore that part of the real world’ discussions.
Right. You don’t even have to leave the solar system. There’s a mirror on the moon now, it gives you a few seconds to work with. Just get those perfect mirrors ready and this project is a go.
Thus the wink, and the implicit nod to your point about perfection.
Yeah, I figured you probably knew, but I wanted to let everyone else in on it, too, because it’s a very common misconception.
Aren’t you forgetting the “perfect vacuum between them” part of it? I know that most of the way between the moon and the earth is a pretty good vacuum - if only we could deal with the pesky last 100km or so…
I figured the other mirror would be on a space craft, but I guess we just need a 100km tube mounted vertically on the ground and we’ll suck the air out of it. These problems are trivial after the perfect mirrors are made.