An IP router has to have a minimum of 2 IP addresses, because its function is deciding on where to send the packet next. A minimal configuration of TCP/IP network where you need a router is a connection between two networks with different network part of the network addresses.
Many times there is also another IP address for management purposes, so you can say that it is the router's own address, but it has nothing to do with the function of router as its merely a convenient access.
So a minimum is one incoming interface and one outgoing interface each of which has to have a different IP address to differentiate them. There can be as much as you want, taking care of many network interconnections, provided you have the needed resources (RAM, CPU, network interfaces, ...).
If there is no choice, there is no routing function so even if you call the box an IP router, it is not. :)