As a big PC gamer I've basically abandoned mobile gaming as it feels like everything out there is trying to get my money by means ranging from insidious to annoying. This is compounded by the fact that browsing the google play games store is a showcase of shamelessness where different developers are releasing essentially identical clones of whatever's popular, it's kind of depressing.
The only mobile games I can bring myself to recommend are Andoku Sudoku, The little crane that could, and crossy road (which even includes monetisation schemes without feeling intensely annoying).
Here's a list of good mobile games off the top of my head. None of these require in-app purchases if they even have them:
You Must Build a Boat,
The Executive,
Out There,
Monument Valley,
Hitman GO,
Lara Croft GO,
DEVICE 6,
The Room (and sequels),
Prune,
Lifeline (this one's great because it would only work in mobile),
Threes!
If you're looking for good mobile recommendations, I'd point you at http://toucharcade.com. They do iOS reviews, but a good chunk of the games also release for Android these days.
I know it seems hopeless but there really are some great mobile games out there which do not try to trick you or screw you around to get your money.
If you like puzzle games (I assume you do, since you mention Andoku Sudoku), you should check out games by Pyrosphere[0]. Especially look at Lazors[1] and The Weaver[2]. Also fun to play is Hoplite[3] and the training mode in Lichess[4]. All of these games are free, though Hoplite lets you pay to unlock an extended endgame, it isn't necessary at all.
I picked up a PS Vita and it's fantastic. So far through various sales I've picked up 35 physical and digital games, so, quite the backlog to last a long while. I don't play any games on my phone anymore. I heartily recommend Persona 4 Golden, Superbeat XONIC, Sparkle Unleashed, OlliOlli, Lumines Electric Symphony, and Mortal Kombat (basically the PS3 version with scaled-down graphics).
The only mobile games I can bring myself to recommend are Andoku Sudoku, The little crane that could, and crossy road (which even includes monetisation schemes without feeling intensely annoying).
Suggestions for others would be appreciated