Geometry Dash. At least, that's what I used to play. flappybird, even though it's not in the store, I still have it with an 186 score. ;)
I have the following on my iPhone: 100 Balls. You basically have 100 balls in a 'box' and there are cups that move around that box. You tap the screen to open a little trap door, which then let's the balls out, in which you try to get in the cups, in which you can then to try and level up. I have other games, but they are stupid... xD
Clash of Clans? If it is okay if it requires internet. Pivot? It's sort of addicting. Solitaire? Good time consuming game. *EDIT* Pivot is no longer free. -_-
Trials Frontier- Good motorcycle game with an interesting campaign. Metal Storm- Great fighter jet game. Real Racing 3- Very fun. All of them include in-game purchases.
My games that I suggest are: Guess the Emoticon (or something like that)- Fun and addicting! Sky Burger- It's fun once you get used to it. Pou- Virtual pet you have to take care of, only downside, turn your phone off, or he will literally kill you with all the notifications. :/ Candy Crush- Addicting. As. Hell. Letterpress- fun and addicting Hero Academy-really fun and great to play!