The 8 wires for ethernet has nothing to do with how many connections you can have. The Ethernet protocol uses those for a single end to end connection (ex: device to switch or switch to switch etc....). 10/100 uses 4 wires while gig uses 8. But that's not really relevant. Some ISPs do have a limit of how many connections you can have at once (often referred to as flows) but it's usually fairly high like in the thousands. These days I'm not sure if it's common for ISPs to have those limits. For mining I really would not worry about it, especially if you are pool mining.
If you're talking about the physical connections, like how many switches you can daisy chain to one, I think there is a theoretical limit but it's very high. You can more or less daisy chain a bunch of 24 port switches to another 24 port switch and then have that connected to yet another switch if you really want to. That's more or less what's done on large networks like schools but it's typically more advanced and may even have redundant paths. It would be using spanning tree protocol. For home you can connect a bunch of cheap switches together fine as long as you follow a tree structure, and don't do anything weird like connecting them in a mesh as it will cause issues.