Link I would say opens up the most so far. That's Smash Bros by default though no unlocks, Mario Kart 8 costume, and a new weapon in Hyrule Warriors for him to use. Most of the rest of the popular Smash Bros open costumes in Mario Kart 8. All amiibo's seems to give bonus weapon unlocks in Hyrule Warriors once a day, which aren't new weapons if you are unfamiliar with how Hyrule Warriors do weapons.
The rest of the games coming up in the immediate future that utilize amiibo's are Yoshi's Wooly World, Mario Party 10, and Captain Toad Treasure Tracker. No clue what they bring yet for those games. We will probably here about Captain Toad soon since that's out on the 5th of December.