I'm going to say that maybe only four ports are active and useful. Some of those home routers have a pair of ports at the end that are either a standard, or an uplink...but you can only use one or the other. Either way, you'll need switches to add as many ports as you need. Unless your router can do multiple subnets, you are probably limited to 254 devices, but you'd need some very expensive switches to get that high.
For the record, having a server or not has nothing to do with your question, as you would still need like network ports to connect all the computers, even if you already had a server.