Occasionally you might need an Autonomous System Number (ASN) that is only used on your network. The need for these should be infrequent, but it does happen.
One thing that comes to mind is a customer that connects to your network in multiple locations. That customer needs an ASN for dynamic routing (BGP, EIGRP, etc.), but doesn't need a registered ASN. Private ASNs to the rescue!
Private ASNs are defined in RFC 1930 and are in the range 64512 through 65535.