The Mogamma
This massive governmental building is located on Tahrir Square and is where tourists go to extend their visas. The visa-extension process is notorious for taking hours, if not days. Some expats avoid the trouble by paying (a very cheap) fine before leaving the country. But I think that the long waits, crowded halls, endless stairs, and commotion of the Mogamma offer a fascinating glimpse into the functionings of Egyptian bureaucracies.