using a set of perfectly aligned mirrors so that the beam itself is visible as it bounces endlessly between them?
Another laser question…
How much would a laser beam diverge on its journey to the moon?
“Endlessly” implies perfect reflectivity (no absorption or scattering) - that’s going to be challenging.
This NASA website says 4 miles:
The answer to the first question is no. Mainly, because no mirror is 100% reflective; the very best we have approach 98-99% reflectivity at a specific wavelength. After about 100 reflections, only approximately 1% of the light remains and this happens within a small fraction of a second.
The answer to the second depends on the beam divergence. If your laser has a divergence of one mrad (milliradian), the beam will have spread to about 380,000 meters–about 1/10 the diameter of the Moon, if I’ve done my math correctly.
But haven’t they used this technique to find an accurate distance to the moon? That must be one powerful laser!
Yes - that’s what the link discusses.
My cat has been trying this for years unsuccesfully.
Gasp
You, sir, owe Kevin T. Amiss and H. Abbot a hefty sum of money for using their patented method of exercising a cat! You should feel ashamed for this!
Exercise? It takes her weeks to get all those mirrors calibrated! 
On the first question, even if you somehow managed to get perfectly reflective mirrors, the beam itself still wouldn’t be visible. For something to be visible, light must be entering your eye, and any light which managed to reach your eye would not be part of the bouncing beam any more.
Also, whenever I’ve thought about this, I’ve always wondered how you would get the beam in to begin with.
What if you pointed the laser through a Bose-Einstien condensate and onto a mirror. Then you quickly close the other face with an opposing mirror. As I understand it, and I don’t, light slows down considerably in a Bose-Einstien condensate. That would give you enough time to close off the other end before the light escaped.
Well, getting off my figurative arse and finding something I read before:
http://science.nasa.gov/headlines/y2002/27mar_stoplight.htm
They have “captured” lasers, just not in the way the OP described (the mirrors can’t have 100% reflectivity).
Sorry the thread title isn’t very helpful, but the answer can be found herein:
In short: no. It’s impossible to capture a laser.
It is possible, however, to create (in a small space) a path that will take so long to complete that it might as well be captured.
Sure.
Put it in your pocket, save it for a rainy day.
You wouldn’t see the beam if it was able to bounce back and forth without some loss.
“Is that a laser beam in your pocket, or…?”
Or until Darth Vader shows up.