Photons are reflecting off the mirror. The image you see when you look at yourself in a mirror is actually light scattered from your face to the mirror and then reflected back into your eyes. Because the mirror is pretty flat and reflects well, it looks like you. A piece of cardboard, on the other hand, sucks at both, so when light from your face hits it a good amount is absorbed and most of the rest of scattered, so you don't see your image.

An important element of this is how smooth the surface is, as this determines scattering. A mirror isn't perfect at reflecting, it's just good enough so that our eyes don't notice, as the imperfections are smaller than the resolution of our eyes. If you scuff the surface of the mirror a little, you'll see your image blur because more light is scattered by the new larger random angles on the surface that scatter the light. But the light still does scatter on a good mirror, just way less.

So, assuming light was traveling back and forth on a mirror, it will still scatter, and that scattering will add up. In addition, some of the light will be absorbed. So, over time, the light will all get absorbed (likely as heat) or will scatter off of the mirrors after going back and forth a bunch of times.

Getting the light to go between them in the first place is actually a bit of a challenge, though. For light to go back and forth for a long time between those mirrors, every time light hits one mirror, it needs to keep bouncing straight back at the other. Nearly all of the light hitting the mirror isn't doing that, though. It's coming from everywhere except the mirror, then maybe bouncing back and forth a couple times, and then off the mirrors, like a zigzag. If you made one of the mirrors a "two-way" mirror you can see through, you'll only see as many "deep" as can travel on that zigzag, which is determined by the size of the mirrors and the distance between them. Let's say they're 2 meter by 2 meter squares 10 meters apart and we only care about thinking about this starting with light from a 1 cm frame. We say, "hey how many times am I gonna see that frame?" That number will be maximized by the angle between the far frame edge and the other mirror edge closest to it. Using trig (hooray) and assuming we need to see the centerline of the frame, that's atan(0.005 m / 10 m), or about 0.029 degrees. Under ideal and slightly synthetic conditions, where we assume you look through the center of your mirror and only have one eye, it'll bounce back and forth (2 / 2)/0.005 = 200 times at most before hitting your eye. You'll see more instances of it at every angle that would evenly subdivides (with an odd number, a point I violated so the math was simpler) the horizontal distance from the center of the mirror to the centerline of the frame. For example, it can bounce 3 times at an angle of about 6.3 degrees. Anyways, if it can go 200 times to get to the center, then after 400 or so it's off the mirror entirely.

So in perfect conditions at normal distances the number of times you'd directly reflect light from the environment is tens to thousands of times. Those framed infinite mirror things are basically the scenario I described and the two mirrors are something like a cm from one another and you only see about 10-30 LED dots before they hit the center.

Please note that mirrors are imperfect, so you will get light that may bounce even more times as it hits slightly random surface imperfections. But that is a very small amount of light, you'd be unable to perceive it.

Finally, consider that mirrors don't emit light themselves, usually. There's virtually no light just going back and forth between them on its own. Just reflected from outside. Also we haven't talked about how light isn't just a particle but can do wave things, but that doesn't really matter for this question. Just wanted to acknowledge it since I keep using metaphors about bouncing back and forth.

This has been your overly long mirror physics adventure with a whole bunch of simplifying assumptions, lol.