Mobile games are designed to be addictive and takes your money, so if that's what you are looking for, get a very addictive game that takes many hours to level up and comes with lots of in app purchases, like Hungry Shark Evolution. They're the type of games where it's basically a second job where you have to grind daily to get anywhere in the game.
I haven't found many good free ones on iOS, usually they are more 'free' on Android while costing a bit more on iOS, that's just how the market is.
Some decent free ones are: Bejeweled Classic, Subway Surfer, and Major Mayhem (this one is very fun).
All of the games by
Mediocre AB are good if you can spend just a bit of money. Granny Smith is one of the best mobile game I've played, it's much better and tons more levels than the $10 Mario Run.
The mobile game I've been playing most consistently for the last few years is
GolfStar, although it's been going downhill with some of the changes. As you level up to Pro status it becomes less about skill and more about spending money or grinding to upgrade your stats and equipment. To me it's more fun to play by yourself than competing with other players, just because of the way they design the whole thing multiplayer and competition isn't fun in this game to me. It is one of the best mobile golf game though.