They do. Portal is a great example. It's not exactly a mirror, but it represents the exact same problem: You're creating a viewport that looks back into the scene, potentially at other viewports which do the same thing.

One of the biggest problems is: Imagine two mirrors facing each other in a room, or two portals facing each other. It creates one long hallway. Portal manages to handle "mirrors looking at mirrors" problem quite elegantly: They render each mirror viewport over and over again down to a depth of 7, and then they put a "good enough" texture on the last one. This ends up looking quite nice since the 8th mirror/portal will usually take up a small part of your screen and you won't be able to tell the difference.

The real question is, why don't most games do this? The answer is some combination of the lack of: time, budget, developers, experience, and bang for buck. In portal it's extremely important to get the mirror effect to look good because you'll be looking at it the whole game. For other games you might run into one or two mirrors ever, so it's not worth it to spend time and effort to get that part of the game to look good when you're already running behind in other areas.

The smart thing would just be to not have mirrors, or to have them be broken/dirty so that you can't see your reflection. With small shards of glass or dirty glass (that you couldn't see your reflection in) you can "fake it" using a simple, pre-baked cubemap effect.