Shattered Pixel Dungeon is a roguelike, well suited to the touch interface and small screen. Offline. Constantly expanded. Free, open source, and on F-Droid.

Developer also appears to have a presence on the Threadiverse, think he came over when Reddit went to hell. Lemme find the community.

EDIT: !pixeldungeon@lemmy.world

https://shatteredpixel.com/

Unciv is a reimplementation of Civilization V for Android. Obviously, less-elaborate graphics, but same gameplay. Free, open-source, available on F-Droid.

https://yairm210.itch.io/unciv

Catacalysm: Dark Days Ahead is an open-world roguelike. The good news is that it is deep, has ridiculous amounts of functionality. Very free-form -- you can build camps with NPCs, mutate your character, acquire bionic implants, construct buildings and vehicles, etc. Some extensive mods to do things like add fantasy content. The bad news is that it also has a very steep learning curve -- think Dwarf Fortress, say. The UI was also designed for a PC, and while the Android port dev did a reasonable job of adapting it for a touchscreen, it's still awkward compared to a keyboard -- not like Shattered Pixel Dungeon. If you're willing to carry a keyboard -- you say that you're okay with a controller, so I assume that you're okay lugging some kind of gear bag -- then it becomes a very good option. There are some folding keyboards aimed at phone use that can be pretty small, certainly smaller than a game controller, if you don't want a more-traditional keyboard. CPU-intensive, though -- in heavily-monster-infested areas, it can load down a PC, and it's probably less-gentle on less-powerful Android devices. Offline. Free, open-source, but nobody has packaged it for F-Droid.

Download links for both the stable and experimental builds here:

https://github.com/CleverRaven/Cataclysm-DDA

https://en.wikipedia.org/wiki/Cataclysm:_Dark_Days_Ahead

Some helpful websites, by decreasing importance (that you might miss if you're playing offline away from an Internet connection):

https://cdda-guide.nornagon.net/

https://www.reddit.com/r/cataclysmdda/

https://cddawiki.chezzo.com/cdda_wiki/index.php?title=Main_Page (often outdated, but also one of the few places trying to aggregate a lot of information from forums and the like).

There's an essentially-inactive community on the Threadiverse at !cataclysmdda@lemmy.world

There is a whole genre of older text-based interactive fiction games that are free and offline for simple virtual machines; the major ones here are glulx, TADS, and Inform/z5. Android has such virtual machine ports; it looks like Fabularium in F-Droid can run them. These involve a lot of typing, as they were designed for the PC, and IMHO are not well-suited to a virtual keyboard, but if you're willing to take a physical keyboard, they can be pretty good. You'll need to learn the (English-like) syntax that the game engines understand. I personally enjoyed Babel and Anchorhead. Two sites that have large collections of free games made by volunteers for download:

Interactive Fiction Archive

Interactive Fiction Database

An intro to the genre:

http://brasslantern.org/beginners/beginnersguide.html

These will be gentle on your battery.

I'll be honest, though -- when I first got an Android device, I was pretty disappointed with the game situation. This is greatly-exacerbated by the fact that I'm not willing to get a Google account and let Google more-readily monitor me, which rules out most commercial games...but I wasn't blown away by even commercial game availability in the Google Play Store, and the open-source situation is kind of sparse compared to Linux, what I'm normally on. Linux is IMHO generally a preferable gaming platform, unless one specifically wants to do touch-based games (which can be important).

I was also kind of disappointed by the lack of choose-your-own-adventure/gamebook-style games on Android. These would avoid the typing in interactive fiction by just having a few choices to select from, which I thought would be a good fit for a touchscreen. There's the large collection of text-based mostly-commercial games at Choice of Games -- you can get their client on itch.io; Android has an itch.io package manager on F-Droid in the form of Mitch that can download it. Heh, though that's downloading a package manager with a package manager to get a package manager. If I had to recommend a few, I'd try Tin Star, maybe Choice of Robots, and the Heroes trilogy; those are commercial, though they have a few free games, and IIRC their client keeps a few normally-commercial games for temporary free play.

While I like the Choice of Games writing, I find that a lot of the gameplay in the games fall flat, more-or-less trying to optimize for playing one character "type" or another; I feel like they're written by novel authors and could benefit a lot from more game elements, and that new authors kind of copied the existing style.

There's a once-commercial series of gamebooks, Lone Wolf, which I can't really call a fantastic example of a gamebook and doesn't have the most-amazing artwork, but which was a real 1980s/1990s series whose author said "go ahead and freely distribute them", so various open-source and commercial projects have gone and done up clients to play the books, do stuff like the dice-rolling and hit-point tracking and so forth. I haven't used Android clients, but they exist. One such project.

https://en.wikipedia.org/wiki/Lone_Wolf_(gamebooks)

I still don't have an open-source solitaire implementation that I'm blown away by, which seems like another surprising limitation. My guess is that you can probably find something non-open-source -- though probably spyware -- on the Google Play Store. PySolFC is on F-Droid. It...works, and it gets me my Eight Off fix (a particular solitaire game that I like but isn't as widely-played as Klondike or Freecell) but it was really designed for desktop computer, and the Android adaptation could be better, IMHO. Small cards and such.

There's a choose-your-own-adventure engine called Twine; games written in various languages -- the most sophisticated such language is SugarCube -- can be converted to Web-based games. That seems like it'd be ideal for Android, and the games are playable on Android, but authors don't always create games that work well on the small screens of many Android devices. I don't know of a single Twine-oriented game archive in the sense that the Interactive Fiction archive and the Interactive Fiction Database serve for interactive fiction games. However, many people who have made Twine games seem to distribute them in packaged form on itch.io. There doesn't seem to be much of an open-source culture around these, unfortunately, so I don't see people doing a lot by creating patches and such. I rarely play these on Android, mostly use the PC. Here's a list of Twine games on itch.io packaged for Android:

https://itch.io/games/made-with-twine/platform-android

There's also a pretty extensive number of adult games for this platform, if that's your cup of tea.

There are emulators for various old game systems for Android. I've used Retroarch on Linux, and it looks like they also have an Android build on F-Droid. I've never spent time using these on Android, because I just always would prefer to play on a desktop platform, but I'd imagine that if what you have is an Android device and using that is a constraint, they're probably fine. That might be the more action-oriented sort of game you're looking for, given that you're talking about a controller. Not much by way of legitimately-free stuff there, though obviously piracy of old console games is widespread, and some people -- such as myself -- will sometimes just buy the game on another platform and conscience assuaged, go pirate it on the platform that we want to play it on. I think my favorite emulated games were probably the most-popular 2D ones on the Super Nintendo, stuff like Super Metroid or and Legend of Zelda 3. Oh, and Castlevania: Symphony of the Night for the PS1. I imagine that a current Android device would have no trouble with any of those, if you've a controller.

Dungeon Crawl: Stone Soup is a traditional roguelike that has a build for Android on F-Droid. This, again, is designed for a PC and is gonna be better-played with a keyboard. It's not beautiful, nor as well-suited to the Android platform as the designed-for-the-platform Shattered Pixel Dungeon. But it has a game that is famous for being refined, with the developer constantly going back and cutting out cruft and grind/busywork, resulting in a very polished game from a gameplay sense. The author, Linley Henzel, has some famous quote about how any action that the player has to make in a game should be an interesting decision, and if it isn't, it should be removed from the game.

https://en.wikipedia.org/wiki/Dungeon_Crawl_Stone_Soup