Hurdles in creating (close to) infinite images Let's put an object (hypothetical superman) inside a "well sealed" box containing only mirrors. Is it possible to create number of images that will be close to infinity, assuming that resolution of our superman's eye is very small (infinitesimal) and he can observe and count all the images that he observes of himself.
If we assume that mirrors will leak some energy, then is it possible to put objects such as a photo multiplier tube (in combination with a mirror) and adjust it in such a way that only the amount of energy lost by reflection of the mirror is recovered and sent back to the other mirror.
What are the other hurdles ? 
 A: mainly that the resolution of the mirror is not infinite, it is made of atoms, so you will not be able to have images smaller than that. But before that, the images will become smaller than the frequency of visible light, so that will also limit the size of the largest possible imagine. Last but not least, unless the mirrors are perfect (impossible), each image will be slightly blurred, and you will get to a point when the blurring makes the original image no longer visible. 
A: 
If we assume that mirrors will leak some energy, then is it possible to put objects such as a photo multiplier tube (in combination with a mirror) and adjust it in such a way that only the amount of energy lost by reflection of the mirror is recovered and sent back to the other mirror.

This cannot be done even as a thought experiment. Photomultipliers are not what you think the name means, and any mass you introduce in your closed box will be absorbing and deflecting light turning it into thermal energy. The body of your superman itself will be absorbing scattered light and turning it into infrared/heat.
As the light is slowly converted to infrared by compton scatterings and absorptions the number of photons available in the visible range to define the images will be falling to the point of single photon reflections, no image,  and finally total reduction to infrared frequencies.
In addition there exists the velocity of light which from the beginning will limit the number of reflected images in a given time interval.
Infinity is a long word.
