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I've got about 160 DS games (all complete in box), about 155 of those are in my backlog :P
While I'm not that sure about the rarity of games - I can only go by the ones that I don't see many of online or are expensive - I'd say my rarest one would probably be EOI, EOII, My World, My Way or Diamond Trust of London Limited Edition (only 1000 ever made!)
Unfortunately it's one of my unplayed games at the moment.
It's a 2-player strategy game about blood diamond trading in Angola - basically a game of misdirection and second-guessing your opponent. It's made by Jason Rohrer who is known for doing lots of small games with interesting concepts - look him up on Wikipedia, it's got links to his stuff.
The reason there are so few copies (6,000 total, including the 1,000 LE copies) is that Rohrer couldn't get funded for publishing by a major company (the game was finished, but he couldn't find a publisher willing to pay for manufacture). He went to Kickstarter to raise the funds to manufacture the game and that's how I got the LE. You can still buy copies of the regular edition from here:
if you're interested - as I write this, there are still about 4,000 copies left.
--I have 60 DS games (US versions) --Maybe about 20 that are on my backlog of which I probably will never get to in the near future --I don't have a lot of "rare" games, that could otherwise be bought online for around 1-2x times the original DS retail price of $30-35. I would say Etrian Odyssey III, (sealed) I guess?
Game Grabs, Trailers, and Pickups! http://www.youtube.com/storino03