To get things going here and also because I'm interested in what kind of fun stuff you're running on your phones, I'll post a list of apps that I'm using daily or that are part of my Android experience.
To start it all off, I'd like to emphasize that I try to run FOSS apps almost exclusively and I also don't use Google Play Services whatsoever (not even MicroG). I do run some very few closed source applications in a separate profile with Insular, but I plain out refuse to run them if they flat out require Google Play Services.
Feel free to tell me what YOU are running in the comments!
My installed Apps
AdAway | Blocks ads by utilizing root access or by acting as a local VPN**
Aegis | Provides 2-factor authentication codes, includes a repository of icons for your services
Aurora Services | Required for background installations with Aurora Store****
Infinity for Reddit | Feature-rich Reddit client (future uncertain due to Reddit's API-pocalypse, however, you can compile Infinity with your own API, as described here)
Insular | Creates a separate (work) profile where you can install "big brother" apps**
Noice | Create and play back profiles of overlapping noises to drown out environmental noises (use version 1.3.3, version 2.x requires internet access and an optional subscription)
SlimSocial for Facebook | Wrapper for Facebook's website, so you can interact with people that refuse to use any kind of messenger, but are still present on Facebook
SuperFreezZ | Freeze applications that you're not using***
Open Camera | Feature rich camera with Camera API v2 support, usually the best FOSS camera on 3rd party ROMs
*CAN be used as a client for self-hosted services
**Offers additional features if root access is present
***Requires root access to be present
****Part of a Magisk module
Edit: Added some links and changed Infinity's description
Here some not yet mentioned #FOSS #Android apps that I think add value to me:
Anki-Droid (github) for spaced repetition learning, synced with the Anki desktop app. I create and manage cards in the desktop app, but a mobile client is convenient for review on the go. Makes memory a choice.
Orgzly outliner using the org-mode markup format of the #Emacs text editor. I use it to view and edit my org-mode notes created in the Emacs text editor on my phone. That use-case is pretty niche, but the app is also decent by itself.
Syncthing for syncing directories between several devices (mobile & desktop) without the need for a central "cloud" server.
Wallabag: Self-hosted pocket alternative. Save websites to read later and sync that list across devices.
Not on F-Droid:
Medito (IzzyDroid, github): Guided #meditation for beginners and a FOSS alternative to headspace and the like. Though if you're experienced just a timer app might be sufficient, there are plenty of those. Might do some tracking, there seem to be some problems getting it into FDROID, see this issue
Thank you very much for your list! I forgot Anki-Droid. I use it to learn Japanese. I used to use Syncthing in the past, but it hasn't been 100% reliable for me. But that was probably a configuration issue on my side.