Piggyback network router
Solution 1:
it's enough if you buy ethernet switch and connect it to one of lan ports of your router.
but yes - you can as well daisy-chain router behind router and have two layers of nat. but then communication between computers linked to first level router and those behind 2nd router will be a bit tricky.
Solution 2:
The simple way, generally not requiring any configuration, is to chain the second router using one of the LAN-side ports. This means you only have 3 ports available, but it's the simple way. There are configuration changes (DNS, DHCP, etc) that need to be changed to do this 'properly'.
Solution 3:
As pQd said, probably the best solution is to buy a 8/16 port Ethernet switch and expand your existing network. If your needs grow again, you can take another port and connect another switch allowing you to expand further.
Just be careful if you connect switch-to-switch that you don't cause a STP loop. See this post for some more details:
What does Spanning Tree Protocol (STP) do?
The risk you will run by connecting one router to another is getting the routing setup properly. Most home 'routers' also act as firewalls which means you will be unable to communicate between the two networks without some additional configuration. You will also need to either run a routing protocol (RIP or OSPF probaby), or setup static routes so everything communicates properly.
Solution 4:
Yes, this is simple to do.
Ignore the WAN port on your secondary router, and plug one of your LAN ports on your main router into one of the LAN ports on your secondary router.
This will bring you up to 6 available ports without the need to reconfigure DNS/DHCP or anything else.