While reading through the great online PHP tutorials of Paul Hudson he said
Perhaps surprisingly, infinite loops can sometimes be h
Monitoring applications
If you have a background process that monitors the state of your servers and sends an email if trouble occurs. It would have an infinite loop to repeatably check the servers (with some pause between iterations.)
Server listening for Clients
If you have a server script that listens to a socket for connections, it will loop infinitely, blocking while waiting for new clients to connect.
Video Games
Games usually have a "game loop" that runs once a frame, indefinitely.
Or... anything else that needs to keep running in the background with periodic checks.