I don't know if the Kindle Fire's store has all this stuff or not, but:
-You Must Build a Boat is a rather unique and fun take on a mashup of RPG, Roguelike, (somewhat; gameplay takes place in a dungon, things are pretty random due to it's nature, and when you die your run is done), and Match 3.
-King of Dragon Pass is a pretty good mobile port, it has some extras in it even.
-Ticket to Ride, Talisman, Asension, Cthulhu Realms, Pathfinder Adventures, Carcassonne, and Catan if you enjoy board or card games.
-Some like the Baldur's Gate games and Icewind Dale offering on mobile, they're the EE versions before Dragonspear came out with the new NPCs put up as in-app purchases. Some hate the interface though.
-The Pinball Arcade will cost you an arm and a leg, but it's worth it if you buy stuff over time. Really great and faithful recreations.
-A slightly less expensive option is Zaccaria Pinball, though the tables weren't as fun to me. Comes to personal taste.
-Bloons TD5 is about the best Tower Defense game I've ever played.
-Riptide GP2 is kinda like putting Wipeout on water with stunts and no powerups.
-Pacman Championship Edition DX is just as good as on PC.
-Dragon Quest IV-VI are pretty much Square-Enix's best porting jobs, if that tickles your fancy. They're the DS remakes all on one screen, with IV restoring party chat that was left out. They also don't have online only DRM. Wait for these to come on sale though, Square-Enix charges higher prices than most. Romancing SaGa 2 and FFIX are also good. From what I've tried the DS FF ports, III and IV, are decent as well as V (though the graphics suck it's ported well otherwise) but I wouldn't go any farther than that. Beware of always-online DRM in I believe FFIV and any games they made after that.
Also I rarely ever get free games, but I've liked Bomber Friends. You can pay for the ads to be shut off and it's multiplayer Bomberman with a Reversi mode. What's not to like? Also, Pixel Dungeon is about the best mobile roguelike if you aren't into getting Hacker's Keyboard installed. It doesn't have ads or anything, it's just free. Battle For Wesnoth's mobile job is also great once you get used to it and set the right resolution. The one by Androth is the one you want.