Of course there are the stalwarts: Infinity Blade (action/RPG), Plants vs. Zombies (tower defense), Angry Birds, Fruit Ninja, Tiny Wings.
I have been playing a lot of My Singing Monsters lately. It's a free-to-play game that is kind of like Farmville or Animal Crossing combined with music. You have monsters, they sing. Each kind of monster plays/sings a different part of the song (one sings a bass part, one plays a drum, one plays a guitar, etc.) Start with a 1-element monster (water), buy another 1-element monster (cold), breed them together and you get a 2-element monster (cold + water). Breed a Level 2 (cold + water) with a Level 1 (stone), get a Level 3 (cold + water + stone). (All of these monsters make different parts of the song) Each monster can be fed to level-up to 15 max. You (the God-like being who breeds and places these monsters) can level-up to 30. Over time, you open up new worlds (new songs), new monsters, etc. Each monster produces a certain $ per minute; 1-element monsters produce just a few, while 4-element monsters produce quite a lot. They also all have a max $ level; 1-element monsters are maxed out at about 5 minutes' worth, while 4-element monsters take 8-12 hours to fill up. (So you have to constantly check throughout the day and "collect" the money from the monsters.) Overall it is just a silly "watch the numbers go up" game, but it is fun because of the music and how you can place the monsters and mute them to customize the song how you want. They are also fun/funny to watch, as they do little movements or dances as they sing. I have paid about $5 in-game to buy stuff that got me ahead, but I've started a second account from scratch and it is really not all that bad to play the game completely free. You are not required to pay real money to achieve anything in the game; everything is able to be purchased with in-game currency, which is earned over time.
Fairway Solitaire is a solitaire game (card game) with a golf theme. It is quite addictive. It is a card game, but at many points it feels like a puzzle game due to the added elements besides the basic cards. I think it is $1 or $2. There are optional in-game purchases as well. It has a large number (150 or so, IIRC) of set "courses" that you can play through, as well as 3 new courses every day that are available for 7 days. A 3-hole course would usually take 3 to 5 minutes to beat; but sometimes there are challenges which require you to play it again in order to get the full star rating for that hole. Again, everything can be purchased using the currency earned in-game.
You probably know of Angry Birds, but the same company makes a game that is really much better called Bad Piggies. You are the pigs, trying to steal the eggs from the birds. In order to do so, you have to build a contraption to get from Point A to Point B on the map. This contraption may involve wheels, engines, propellers, wings, helicopter rotors, etc. You can experiment and build the contraption any way you want, and often there are multiple ways to solve each level.
