Urban Fantasy, High Fantasy, Epic Quest, Sci-Fi, Time Travel and World Hopping, Steampunk, Action/Adventure, Modern Drama, Mystery, Slice of Life, Romance, and many more.
Splashy Fish (an addicting side-scroller game that is simple and (you'd think) easy)
Frotz (A choose-your-own-adventure type game where the story responds to what you type in the breaks. I highly recommend Lost Pig which comes with it for free)
Clumsy Ninja (help to train a ninja in all kinds of goofy ways so he can rescue his sister)
Flow Free (connect the pipes, and make sure to use up all the space! Trickier than it looks)
Chibi Me (how I got my avatar; you have to pay for more options, but it can be fun to play with)
Brave Frontier (Occupying if you want a more console-style RPG to play. Not much story to it though, it's pretty simplistic)
Despicable Me: Minion Rush (addicting as all hell)
Bejeweled
Hangman by Spice
SCRABBLE free (only play if you're ok with the use of slang, proper nouns, and nonsense words in scrabble)
Can You Escape (a look-around-the-room-collect-items-and-solve-puzzles game!)
Falldown! 2 (guide the ball through falling platforms; pretty addicting and you can unlock new balls and backgrounds)
Sonic Dash (Reccomend to any sonic fan; very fun)
Guitar! By Smule (Guitar Hero for your phone, basically. Limited but growing selection)
Draw Something free (prepare for penis pictures if you play with strangers)
Tetris Blitz
Plague Inc (manipulate a virus and exterminate humanity with it! Very fun and challenging)
You Rather (fun, and a decent conversation starter)
NinJump (also fun, simple, and addictive)
Sound Key (I highly recommend paying the $2 for the full version, but the free version is pleasant, too. Touch the notes and make music)
Mangastorm (very useful if you're a manga fan, best with an internet connection)
CameraTimer (there's popups, but if you like to take good pictures this helps you take multiple shots and choose the best one, and take pics without the arm-in-the-picture style selfie shot)
PokeMaster (handy if you're a competetive player or a breeder)
MultiDie, MyCountR, Gasbuddy, Ebay, Google, and my personal bank app so I can do quick transfers and balance checks, are not entertaining, but are very useful