This is a screenshot of the "games" section of my phone's Home screen. (It's massive, so don't click it unless you really want a 1440x2560 image in your browser.) Here's what I've got in order.
HUEBRIX - Puzzle game (paid). It's sort of like those "Flow" games, where you're supposed to drag a line from a source that goes exactly a certain distance (indicated by its number) without crossing paths or leaving any blank spaces. This one adds a bunch of obstacles like one-way tiles and stoppers. Cool music, too, sort of Tangerine Dream-ish. Only thing I don't like about it is that the game makes fun of you when you finish puzzles, even if you do well at them ("Impressive, for you.")
The Room Two - Sequel to The Room and NOT related to the Tommy Wiseau movie at all. It's a Myst-like puzzle game where you're trying to unlock mechanical puzzle boxes. Sometimes it stretches the bounds of physics, but it's really rewarding to see the puzzle box moving when you've solved it, and the sound design in this game is amazing.
rymdkapsel - Combination puzzle game and real-time strategy, you have to manage a few different kinds of abstract resources while expanding your base and defending it from weird flying monsters. It would work better on a tablet, I think.
Pixel Dungeon - Be careful when looking this up on Google Play; its open-source nature means that there's about a hundred different clones of it, some demanding in-app purchases, others demanding an up-front fee. The original is ad-free and costs nothing. Great roguelike game that adapts well to the phone interface.
Pocket Tanks - Haven't paid money for the deluxe version yet, but this is the best Scorched Earth clone I've played, from developer Michael Welch, who also made the popular Amiga shareware game Scorched Tanks. Has pass-the-phone multiplayer, and the only limit of the "free" version is the number of available weapons that the game randomly picks from.
Canabalt HD - Arguably, the only good infinite-runner game, the Android port adds "two-player" mode and some special challenge levels as well as sharpening up the graphics a little (you can go back to the originals if you want). In a pretty awesome gesture, there are buttons in the app to dump the in-game music into your Ringtones folder, though honestly only "RUN!!" really works as a ringer.
250+ Solitaire - It bills itself as "free" but has semi-intrusive advertising unless you pay three bucks in the app. I did that. There are lots of solitaires in here, with full instructions and demonstrations to watch, the interface isn't a pile of shit like a bunch of other solitaire packs, and if 250+ games isn't enough, there's even a simple game creator included so you can throw together your own custom variants.
Raiden Legacy - Not the greatest ports of Raiden (it includes original Raiden and the three Raiden Fighters games), but it works well enough. The new interface slightly unbalances the game in places, since you can move your ship as fast as you want by dragging a finger across the screen, even if it's a really slow ship (like the Raiden Mk.II in Fighters Jet). Still, looks and runs great, and aside from some awful sound emulation in Raiden 1, sounds alright too.
Does Not Commute - Highly amusing "racing" game where your goal is to drive a car from one end of the map to a predetermined exit. Thing is, every time you do this, the last car you drove is saved, and you have to navigate the next car around the last one. Every car on the screen is you. You are your own worst enemy. Great music and some hilarious writing accompanies a fast, simple game.
spelltower - Word game, similar to Bookworm, which is great because PopCap Games has NOTHING on the Android store.
Simon Tatham's Puzzles - Solid Android port of an open-source collection of simple puzzles, from nonograms to Netwalk, minesweeper, SameGame, and a lot of pen-and-paper adaptations including a reasonably solid implementation of Sudoku.
Dice & Hexagons - A clone of the Flash game, Dice Wars. I'm honestly not wild about this one, because it makes you unlock a lot of the game settings and intersperses each match with ads, with no clear way to get rid of them.
Karateka (Classic) - Solid emulation of the original Apple II game by Jordan Mechner. Touchpad controls are slightly awkward but workable, given the game's relatively slow pace. There's an unlockable rewind option once you clear the first area, and there are options to enable/disable scanlines or green/amberscale monitor. And yes, you can still walk backwards off the cliff at the start and kill yourself.
Hundreds - This one is a little too tricky for me sometimes. Neat soundtrack, though.
Ending - A puzzle game that styles itself after classic Roguelikes, your goal is to get to the exit in a limited number of turns. You can get more turns by killing things (by touching them), but they can just as easily kill you if you accidentally run into their kill zones.
Try it out in Flash here; only the mobile ports cost money, and it's only a buck anyway.
Super Secret Service - Simple arcade game with great pixel art and SNES-style music. You control a pair of on-screen Secret Service agents and must dive in front of the President to block objects that are thrown at him. Kinda tricky, since if you miss a dive, you have to wait a second or so before you can dive again, so you have to hold back that itchy trigger finger.
Time Surfer - Got this from a Humble Bundle a while ago; it's a neat endless-runner with an 80s-neon flair, from the guy that made Bean's Quest. (You can even play as Bean if you own that one.)
Coffee Mahjong - One of those Amazon Free App of the Day things, it's basically just like the old Mahjong/Shanghai type puzzle games, except the tiles have coffee, donuts, and spoons on them. Nothing mindblowing, but it's better than most Mahjong clones out there.
Sticks - Haven't played it yet.
10000000 - Because Android doesn't have the original Puzzle Quest. Action-puzzler with RPG elements and...ugh...fake-retro graphics. Cool music, though, and the game's fun enough that I care less about the visual style.
Sigils of Elohim - Polyomino puzzle (fit all the tetris pieces into the grid with no overlaps) that ties into Croteam's The Talos Principle. Simple, effective, great for pee breaks or whatever.
FFXIV Libra Eorzea - Pretty much just an official wiki for Final Fantasy XIV: A Realm Reborn; lets you look at your character and search for items, monsters, and NPCs. The fan-run wikis are better (especially Gamerescape's).