Does this router exist?

You can always daisy chain switches into your existing router. That's why at the consumer level, you're not going to find many w/ more than 4 ports. They’re trying to keep costs down as much as possible. 90% of ppl will probably never even use those 4 ports. Even the BEST routers at the consumer level (D-Link DIR-655, D-Link DGL-4500, etc) are typically 4 ports.

If you let the number of available ports drive your decision, then you’ll probably end up w/ a VERY small selection to choose from if you want something reasonably priced. And that just doesn’t make sense. Never let something as trivial as the port count limit your routing options.

JMTC

Something to consider: an actual "ROUTER" only has 2 ports. The "routers" you see with 4 or more ports actually have an integrated "switch" to provide the extra ports. The cost effective thing to do is look for the right router you want, and add an 8 port switch so that you'll have all the ports you need.