You have received a number of responses on this, most correct in whole or in part, however none have fully answered your question, and some have given what I consider less than optimal answers.
First of all there is a big difference between a physical port and an IP address. It is quite possible for a single port on a single host to have multiple IP addresses assigned to it, but any given IP address must be assigned to no more than one host on any network. So the answer is, yes, your second TiVo will require both a second Ethernet port and a second IP address distinct from the first. As one poster mentioned, one can daisy-chain multiple switches off a single router port, and while there is no hard limit to the number of switches in cascade, I recommend you don't endlessly cascade one switch after another after another. A much better practice is to purchase a larger primary switch to handle more traffic and then hang at most one switch off each port of the primary switch. With 24 port switches being very economical these days, a single primary switch with a handful of subtended switches could easily take your LAN to more than 128 hosts, at which point I suggest you invest in a router which can handle multiple networks. I take it, however, you are still a long way from that end of the roadway. If you have used up all the ports on a 4 port broadband router, however, then it's not unlikely you may eventually fill up another 4 or more. Therefore unless money is really phenominally tight (unlikey for someone buying two S3s) I suggest you get a somewhat larger switch - at least a 12 or 16 port switch. With the advent of fairly inexpensive managed switches, you might even consider getting a managed switch. It's more money, but it may stand you in good stead in the long run. Of course, managed switches cost significantly more than unmanaged switches, but there are a number of decent managed 16 and 24 Port Gigabit switches available for between $200 and $500. If this is truly overkill for you, then I at least suggest a 16 port unmanaged Gigabit switch. At about $50, the extra investment is minimal, and it will stand you in good stead for a while to come.
One of the posters suggested a wireless solution, but unless a wired solution is not practical, I suggest you stay away from wireless solutions. They generally provide much lower bandwidth at a much greater cost, are prone to various sometimes subtle and difficult problems, and security can be a bit of an issue - especially if the user is not familiar with wireless security protocols. Although not terribly difficult, if the user is not capable of handing a managed switch, then I don't recommend they try managing a wireless network, either.
Personally, I'm using a TrendNet 24 port Gigbit Web-smart switch. It cost just a bit less than $400, and it's served me very well when troubleshooting LAN issues (or proving the LAN was not the issue). It currently has 18 ports in use. I can transfer files between workstations at better than 220Mbps.