Back in the day, the 'Random Game' was actually a random game, I think, but it made the site lag too much, so they pulled it. They might be implementing it in AG3... I forget. If you look on the beta site at the moment, they do have a 'Promoted Game', though.
If you want to do it manually, though, you could program a random number generator between zero and the latest game number (the latest one says 13160). You'd have to do some reloading, though, since there are about 2,300 some games between the numbers of 0 and 13,161. You just type 'http://armorgames.com/play/' and the random number, and that should take you to a game, if it exists.
