F-Droid/OSS Apps

[DarkMahesvara]

I personally like the following

Messenger:
Session - decentralized E2EE messenger (fork of Signal)
Signal - centralized E2EE messenger (not 100% OSS/has proprietary google components but there is a FOSS fork)
Element (previously Riot.im) - Secure decentralised chat & VoIP

System:
PCAPdroid - Monitor and record all Network traffic (without root)
App Manager - See App info (permission, tracker and much more)
Exodus - Show and lookup trackers and permissions of installed apps
Smarter Wi-Fi Manager - Set WiFi state depending on nearby cell towers
Orbot - Route traffic through TOR
Haven - Record nearby activity with your devices camera, microphone and sensors

Security:
KeePassDX - Password manager compatible with KeePass files
andOTP - 2FA TOTP authenticator

Media:
NewPipe - Lightweight YouTube frontend. No Ads or Tracking and everything is stored locally
UntrackMe - Transform Youtube, Twitter & Instagram links to their free and open source alternatives

Traveling:
OsmAnd - Global Offline and Online Map Viewing & Navigation

Games:
SuperTuxKart - Mario Kart but FOSS
Anuto TD - Simple Tower Defense
Freedoom - Doom engine source port bundled with Freedoom 1 & 2
Shattered Pixel Dungeon - A roguelike game based on Pixel Dungeon

Miscellaneous:
Aurora Store - download Playstore apps
SecScanQR - QR code scanner/generator
Survival - Manual Learn how to survive
Kakugo - Learn Japanese

 

[Jahy]

I'd like to add to this list by mentioning Tachiyomi, which is without a doubt the best way to search, download, and read manga and doujinshi on a mobile device. It allows for a great deal of customization, both visually and functionally, and allows you to utilize a wide variety of sources, most notably MangaDex which is where most people should be getting their translations anyway. Also, if you're a tracking nerd like me, it also seamlessly integrates into your MyAnimeList, AniList, Kitsu, etc.