Any Monster Hunter game (Monster Hunter 3 Ultimate/Monster Hunter 4 Ultimate recommended for newcomers/solo players).
Any Pokemon game (Pokemon OmegaRuby/AlphaSapphire recommended, most content and best mechanics).
Any Fire Emblem game (Fire Emblem Awakening recommended to start)
Bravely Default (a traditional turn-based rpg)
Zelda A Link Between Worlds (other 3DS Zelda games are just remakes, this one is brand new made for the 3DS)
And I'd even suggest looking into the world of homebrew/custom firmware. Let's you do all sorts of cool things on the 3DS one never thought possible. Like using a normal controller connected to PC to play on your 3DS (makes games like Smash Bros and Monster Hunter more playable) or being able to stream the video output to a bigger screen (for livestreaming or playing on a bigger screen, used to require sending the 3DS off for a hardware mod and paying hundreds of quid).