One is you can connect a modem for dial-in/out. ODR is now an option, at very low speed, but enough to connect to the router to fix any problems. You can also use it to download an IOS image to the device. There are probably more.
Another possible use for the aux port: I answered a post in NetPro a while back where someone was having trouble accessing their router via the console. It turns out that someone had configured (by mistake) under line con 0 the command "no exec" which stops the console from accepting input. They were able to access the router via the aux port and change the configuration of con 0 and restore access via the console.
