If we're talking mobile games for phones, I know off the top of my head the best non-super heavy microtransaction ones are The Walking Dead No-Man's Land, a zombie XCOM-like with vague Clash of Clans like outpost building. Quite good, just play a couple missions a 'tick' and take it slow otherwise you run out of your 'gas' which is parlance for 'timegate'.
Everyone knows about Fallout Shelter but here I go again; Fallout Shelter's a very decent little builder/management game that's also got a ProgressQuest like side game where you send vault dwellers out into the wastes to come back with random shit.
D already posted about the Trese Brothers, good board games for decent prices.
Minecraft Pocket Edition is pretty good, if you want Minecraft on the go. It's a little behind the 'main' game but it does get occasional updates to bring it closer to the main version.
There's always phone Agar.io if you wanna accidentally run into things bigger than you and die cuz you couldn't react fast enough using touch screens.
Warhammer 40k Space Wolf is a turn-based tactical strategy game that uses a card mechanic. The cards can be something like 'sprint' or it could include any weapon so occasionally your single tactical marine will somehow lug a rocket launcher, flamethrower, and heavy bolter all at once.
Star Wars: Uprising is a sort of mini-MMO that was vaguely appealing until I had to delete it for space. I didn't get too far, but it's basically a mission based RPG.
And that's my list of ok mobile games. What I really wish is for a good mobile sort of Magic the Gathering type game. Yeah I know, Hearthstone, but still.