I’ve been playing Magnibox on Steam, but it just recently come out on iOS and Android. It’s a sharp little puzzle game with well-designed levels and a smooth difficulty curve. The trailer here does a good job showing off the mechanics, but in short you need to roll a box to the goal. One side of the box is an electromagnet you can activate to interact with whatever it’s facing: zip towards or away from stationary blocks (depending on polarity), pull or push movable blocks (again, polarity), and more. It’s a good puzzling time, no ads, one up-front payment.
DATA WING (iOS, Android) is a completely free arcade racing game with a story about AI. Controls are simple: you’re constantly moving forward, touch either side of the screen to turn that way. But it feels great to play and there’s enough control nuance to keep you going especially if you’re a time-attack minded player. You get a speed boost from thrusting away from the course wall and finding a good racing line feels like finding a good grind line in Tony Hawk.
Hoplite (iOS, Android) is an elegant tactical puzzle-roguelike with simple rules and fully deterministic combat. It’s all about smart movements and careful trade-offs. You’ve got a leap ability that skips over one tile, but it costs energy. You can throw your spear, but then you can’t stab anyone until you go pick it up. iOS version has a low up-front cost, Android is free with a premium upgrade to get more available perks and a challenge mode.
Twinfold was already mentioned by mnemos, but Kenny Sun has made some smart games before that. Like Yankai’s Peak (iOS, Android), a mind-bending Sokoban with triangles, and Circa Infinity (iOS, Android), a platformer delving deeper into concentric circles. All his games are also available on PC and have low up-front costs.
I assume most people have played Downwell (iOS, Android) on some device, but it’s still one of the best roguelike platformers and it was built for phones. It has a low up-front cost on both platforms.
Vextor is Android-exclusive as far as I can tell, but it’s a lovely translation of Geometry Wars style twin-stick shooting for phones, and it’s free. Cool feature: landscape orientation is twin-stick, portrait is single-stick with auto-aim for one-handed play.
Simon Tatham’s Puzzles have been ported to iOS and Android, and they’re still great free implementations of dozens of classic puzzles.
Smartphones are a great fit for text adventures. I don’t know which interpreter is good for iOS, but on Android I like Fabularium. It supports all the major formats, of course, and it’s free.