It depends on what you like. I'd recommend New Super Mario Bros, just because hey, it's Super Mario Bros!
If you like RPGs, pick up Final Fantasy 3 & 4, because they're both really, really good. I haven't played it, but I've heard nothing but good things about The World Ends With You.
Also, I don't know how good your Japanese is, but if you like horror games, Square Enix's "Nanashi no geemu" is really, really good and quite scary (but Japanese only).