Ingress is a blast if you like AR games and geo-location based gameplay. I don't know if it is still invite only but I am willing to re-install it to send you an invite if it interests you.
Mahjong Artifacts is a really clever and fun take on solitaire Mahjong. Best I have ever played. I will eventually buy the sequel as well.
The Sea Empire games by Blue Plop are all great strategy games. The first two are free with minimal ads. Third is more resource intensive.
Star Colonies is also by Blue Plop and plays a lot like those classic 4X Strategy games. Only the demo is free but still a nice game.
Empire Defense 2 is a very neat tower defense game. It is entirely free though you can pay to get some things faster (but if you learn how to play right it isn't too hard to earn crystals). It also has a fun online competitive raiding/ranking system.
Alchemy is more of a time waster but fun for what it is: combine elements to make new elements and unlock all of the different combos you can.
The traditional board and card games by AI Factory are very well put together and can include local multiplayer.
250+ Solitaire Collection by Alexei is hands down the best solitaire collection game. Though the most recent update includes some form of micro transactions (I think you can buy new backgrounds and card designs) so I have not updated that yet.
Carcassonne is a digital version of the board game. It is how I learned of the game and is very fun. Has local multiplayer.
Castle Doombad is a fun and entertaining tower defense game. It is similar to the WiiWare Final Fantasy: My Life as a Dark Lord where you play the villain in his keep keeping out heroes.
All of the Cut The Rope games are fun, cute, and charming puzzle games.
The Cut the Rope team made another fun puzzler called Pudding Monsters. This one is more of a sliding puzzle to get all of the monsters combined without losing one. I prefer Cut the Rope but it is different enough to mention (not as difficult).
Where is My Water? Is also a fun puzzle game.
World of Goo is on phones now and if you have not played it, you simply must. Best Game On WiiWare. Period. (Was my GOTY when it released).
Heroes: A Grail Quest is a delightful Strategy RPG that feels similar to Ogre Battle.
I just got Star Traders RGP and it plays like those old DOS games where you travel space and explore, trade, and fight aliens. It is also a call-back to Sid Meyer's Pirates (but in SPACE!).
The Airport Mania games are a blast. They have you managing the incoming and outgoing planes in an airport. You can upgrade the terminals and supply pumps with money made from successful plane deployment.
Gem Miner is a poor-man's Steam World Dig. Intriguing but did not play long.
Greedy Spiders is another charming puzzle game where you try to save bugs from an approaching spider by cutting strands of the web.
Rebuild is a surprisingly engaging turn-based strategy game where you become the leader of a group of survivors in a zombie apocalypse. You send different team members out to explore and gather resources, handle surprise attacks, equip and level up team members, deal with moral survival dilemmas, and work on reclaiming territory in the city. There are a lot of ways to win including finding a cure, purging the city, running away to be a hermit, etc. Like I said, surprisingly engrossing. I would think of it as a sort of "Sim-Zombie-Apocalypse".
Starlink is a very fun, simple strategy game. You have constantly spawning units which you send to take other planets, boosting your production rate. You race against other players to take the most systems and steal theirs by force. Clever deployment of your armies with assistance from special powers helps you win.
Plants Vs Zombies 1 is a must play tower defense game. The sequel is fun and for the most part playable, but you can see how EA has tainted the series. I played the free version until an update ruined my save. Now I am holding out for a 3DS or console version that is a "complete package" instead of including micro transactions.
ANYWAYS
I have played others but these are the ones most worth checking out. Ask me about any of these and I would be happy to expand my thoughts. Not all of these are free. Be sure to get the Amazon App Store as they give you a free app of the day (sometimes garbage, read the reviews before you download them). It is how I got all of my paid apps (never paid a dime for a phone game/app). The ones I listed are ones I would pay for now that I have played them, though your tastes may vary.