If you're into deck building give "Dominion" a go. The base game is free (which is absolutely playable) and a good way to get into the swing of the game. The AI is a cheating bastard through, so either play it on "easy" or play with some friends. It's SERIOUSLY addictive.

I've been playing old electron engine game like icewind Dale & Baldur's gate. Even Neverwinter nights runs & plays great.

Chess.

Hasn't gotten an update in awhile but the community is thriving.

Game Dev Story was a lot of fun.

baba is you is pretty good.

Most of the mobile games I play have been mentioned by others, but I'll give a shout to one I haven't seen in the thread: Dungeons of Dreadrock. A fun puzzle game with a good amount of content. Free with ads, or ad-free with a onetime purchase; no MTX. Doesn't offer the same replayability as something like Shattered Pixel Dungeon, but I was happy with what I got.

Backpack Battles

Very fun strategic game, there are MTX in the game, but they're not necessary to play in the slightest.

I have bought a few things to support the Devs though.

Slay the Spire

The Battle of Polytopia is a fantastic turn based strategy that I've spent way too much time playing.

Beholder is also great on mobile. Similar authoritarian setting to Paper's Please, lot's of decisions, consequences and replayability.

I Love Hue is a super relaxing colour puzzle game.

Outside of emulators and other similar apps, there are some games that have decent mobile ports. Some of the good ones I know of are:

Lego Star wars: the Complete Saga, which as far as I can tell, seems mostly identical to at least most of the other versions but it lacks all of the co-op/multiplayer components. I did also see some minor graphical glitches but they didn't effect gameplay.

The 3D remake of Final fantasy 3. It seems identical to the PSP version but it's obviously running at a higher resolution and possibly a higher frame rate as well.

Bloons TD 5, other users have mentioned 6 but 5 is also good too.

HamSandwich, which is not available through the Google Play Store, so you may soon be stuck with the browser version when Google's changes to side loading take effect. Also, I recommend using either a keyboard or a controller, as the touchscreen controls are not that good.

The Enchanted Cave 2, the first one is also pretty good too but 2 has a lot more content and is a much better game all around.

I know that a lot of people have issues with games that have MTX, but I will defend these games if they are actually good and I can play them without feeling like I'm being force to put money in. Here are some of the games I can remember:

Honkai Impact 3rd, I haven't completed the story yet but I've played up to the end of chapter 28 of part 1 and I haven't had to put a single cent into the game. My only issue with this game is that, while the game does have keyboard support and the keyboard controls are decent, it doesn't have controller support for some reason.

Bloons Monkey City, this game is basically just Bloons TD 5 but with a city building mechanic and some online components.

The Walking Zombie 2: Shooter. This ones a bit weird but I've been playing this occasionally. It's a pretty decent FPS that takes at least some inspiration from the Fallout games. It's not a masterpiece by any means but it still seems worth mentioning. I just wish it had keyboard and mouse support.

There are many great Indie Games. My far favourite are the ones by 'yiotro' which seems to be a single (Russian?Ukrainian?) developer. Each is some kind of strategy/simulation mix and very cheap with simple graphics but a lot of depth. I discovered his work from seeing Antiyoy which either is so old that I got a demo of it in a ~2002 gaming magazine, or a copy of that old game. Just amazing fun all around and the only reason I don't have any of them on my phone right now is that I can't stop playing if I install them (similar to Unciv and Shattered Pixel Dungeon, both FOSS and excellent choices that were mentioned already).

Tetris.

Maybe it's not what you're looking for, but classics are classics for a reason. If you're on anything remotely standard you can find a port.

https://sudoku.coach/ is installable as a progressive web app and is probably the best sudoku app I’ve encountered. Extremely customizable and good for learning new solving techniques

Forkyz is a crossword app that is open source. I love it.

Shattered Pixel Dungeon is quite a timekiller on the phone

I think no one mentioned these yet: My favorite tablet games are the ports of old BioWare classics like Baldur's Gate 1+2, Icewind Dale, ... Excellent ports imo, bur I did play them on a 10" tablet, I wouldn't try to play them on a phone. https://www.beamdog.com/games/

https://gitlab.com/lecarore/breakout71

Some great suggestions in here, but I'd also like to mention This War of Mine. You can read all you like about how good and gut-wrenching it is, but I still always need to schedule some light downtime after playing it for a bit.

Alto's Adventure. One of those simple is powerful games. Though it can get really frustrating as there definitely are situations where the game just decides that actually, you're done.

Old school RuneScape

I Love Hue

It's a casual color puzzle game. Relaxing and minimally intrusive

Unnamed space idle: free idle game without any ads kr necessary purchases

Besides the waiting, Clash of Clans is actually decent.

(And besides a few youth sins) I am 100% free to play)

In my experience the play store is awful to use if you're looking for anything that's not full of ads and/or extremely monetized. I recommend only using it when you know exactly which app you're looking for.

Generally I find games through itch.io and recommendations from others. Also there are foss game emulators on GitHub and similar sites.

Netflix recently started offering mobile games as well.

Ages ago humble bundle did android games but that's all in the past now.

I've been enjoying worldbox and lichess. I used to have a bunch more but the latest Android version made most of my collection stop working.

One thing I don't see people mentioning is that Netflix has a bunch of good games for "free" if you already subscribe.

You Must Build a Boat is an RPG/match 3 hybrid with no shady shit. Pay once. Own the game. It's pretty fun.