At ExtremeTech, John F. de Perczel has transformed an old NES console into a home theater PC using VIA's mini-ITX form factor EPIA motherboard and a 1GHz C3 processor. He says, "it streams music, surfs the Web, downloads movies, and runs NESticle like a champ. It won't, however, run iTunes visualizer in full-screen mode."
The page is dated 8/25/04, so we're willing to bet there are plenty of more powerful options available now if one were interested in updating the concept. How about shoving one of those new Mac minis into an NES?
Check out the Nintendo PC slideshow.
NES alarm clock; hit Reset to sleep in
UPDATE: Ask and ye shall receive! Mac mini in an NES.