Of course, if you want to get technical, "Cops and Robbers" is a form of LARP. So is "playing house" and all the other make-believe games that kids play. Which reminds me of an essay I read in one of my Changeling books...
Steve Herman writes in an article entitled "Oh Boy! A Cat's Eye Shooter!" from the Changeling Player's Guide about how the imagination, the firm belief in things wondrous and magical, is the most precious gift that people have, and the loss of that belief as we grow to adulthood is a terrible tragedy. The author describes how he and his fellow gamers got out their old toys, action figures and the like. They tried to play with them, as they had when they were younger. It wasn't fun for them. They couldn't "create jungles from houseplants or deserts from carpet."
So they returned to the table. They played Changeling; they found that wonder again. They tried to steal a Wayne Gretsky rookie card, and chased a purple Snozzwanger through the alleys.