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
Thank you, Obtanium is a great piece of software, too.
For those that are going to use it, though... Keep in mind that only applications that are on F-Droid are sure to be free of proprietary libraries, etc. I personally tend to stretch the rules a bit, so that's fine with me. But there are some Richard-Stallman-esque people who truly only want FOSS on their phone and I get that.
Also, if you installed something from F-Droid (or sometimes from somewhere else) and then try to update the same app through obtanium, it'll refuse to work, unless you uninstall the already installed version first. Same goes for updating F-Droid apps through Aurora Store (or Play Store).