What's stopping an infinite number of altcoins with the same protocol from appearing in the future? My novice understanding of BTC is that BTC is valuable because it's scarce - mathematically, there can only be 21M BTC, no more.  

What's stopping another altcoins from starting with the same protocol, but a new chain, rendering the limit of BTC to N * 21MBTC?
Nothing stops this at all.  You can start your own right now, removing the 21M limit or adopting whatever rules you want, and mine as many coins as you like.

Likewise, you can sit down at your printer, and print lots of $100 bills with your own portrait, and invent whatever rules you want as to how they can be spent.

The question in both cases is whether anyone besides you will want to use them, and be willing to exchange them for goods or services.  You'd have to convince people to use them, that your rules are such an improvement over the existing ones that they should switch.   If they don't, then your new coins or bills have no real value.

So there's nothing stopping an infinite variety of altcoins from appearing; but probably only a few of them will be adopted by anyone.