I'll second Ghost Trick. It's developed by the same guy who did the Phoenix Wright games, and it's his best work by far.
999 is good if you're not bothered by bland over-explanatory anime writing and lousy route structure. Play it, but get a darn no-spoiler guide (solve the puzzles yourself, though), otherwise you'll get tired of the game before you even get to the true ending.
Devil Survivor is good as well, but you probably want to grab the 3DS port since it has extra content. Devil Survivor 2 has improved gameplay, but the plot and writing took the anime feel of the original and turned it to eleven. You pretty much have to turn off your brain to enjoy that aspect of the game.
Art academy if you want to develop bad drawing habits.
I'd recommend the DS Advance Wars, but I haven't played it. It's Advance Wars though, so it's most likely good.
Henry Hatsworth is great and you should get it.
Elite Beat Agents. Pretty much the only good Rhythm game on the DS.
And I'll probably think of something else later.
Edited by Waller, 08 September 2014 - 05:41 PM.