I've been using plenty of neat apps from F-Droid, here's a few that I think are worth sharing.
Droid-ify - A better F-Droid frontend, best to start with that.
Fossify Suite - After SimpleMobileTools got bought out by ZipoApps, one of the maintainers forked it and started the Fossify project to continue the development. I use Gallery, Phone, Contacts, SMS and Calendar, so all the basics and it's working great. Contacts and calendar are synced with my home CalDAV server via
DAVx5.
KISS Launcher - Simple, functional app launcher. A searchable list that can also look up contacts and app shortcuts, remembers what you've opened the most and can be customized to be as minimal as possible. In my case it shows nothing but the wallpaper on unlock. No need for more.
Unexpected Keyboard - A very simple keyboard that's more oriented around PC users with a layout that's very close to your typical ANSI keyboard, the ability to use Ctrl, Alt, Tab and all the typical PC keys and related shortcuts like Ctrl+C, Ctrl+V, arrow keys, and to type in alt characters/accents you just swipe to the corner of the key, fast, intuitive, and is better suited for your PC keyboard muscle memory. No autosuggestion, no swipe typing, you gotta rawdog it just like you do on PC.
URLCheck - A URL opening intermediary that can redirect links to different apps, clean them from tracking strings, unshorten URL's, do VirusTotal scans and other neat things.
Scrambled Exif - A sharing intermediary for cleaning EXIF data. So before you share a photo to another app, first share it to Scrambled Exif, then share it again to where you want it to go. Simple tool for that extra bit of privacy.
K-9 Mail - An e-mail client that's part of the Thunderbird project and is bound to be rebranded to Thunderbird for Android once it hits feature parity with the desktop version. It works, it's good, it supports multiple account and has a unified inbox.
OsmAnd~ - Offline maps and navigation based on OpenStreetMaps. The Google Play version is paid, but the project is open source and this is the legal free version of it compiled from source. Has all the features of the paid version, handy to have in case you end up outside of mobile range but still well within the GPS range.
PySolFC - Mandatory time waster, right now the only game I have on my phone.
There are also some apps that I use that aren't on the main F-Droid repo, or are only available on Google Play and some are paid, but they're worth the price.
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Tasker - An incredibly robust app that can automate tasks or add extra functionality to your phone, like a customizable dashboard screen, automatic VPN connection based on specific conditions, special shortcuts to specific menus or options, and really, anything you might think you'd want to do, you can probably do it with it. Paying extra for AutoInput, AutoNotification and AutoTools plugins is also highly recommended.
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Edge Gestures - Samsung as One Hand Operations + for their phones. This is the next best thing for every other Android phone. Once you customize it a bit it'll improve your phone's usability tenfold. You can assign Tasker shortcuts so sky is the limit in terms of how you want to adjust it to your needs.
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Poweramp and
Poweramp Equalizer - If you're a music nut these are a must have. Poweramp is undoubtedly the best local music player for Android and with Poweramp Equalizer you can use the same robust equalization for other music apps.
Total Commander - Completely free on Android and just as powerful on it as it is on Windows. I recommend using the F-Droid repo versions as the GPlay ones had to be gimped to comply with their rules. The extra plugins for things like FTP, SMB or even cloud providers are also very, very useful.
GrayJay - If you've heard of Louis Rossmann then you've surely heard of this. It replaced YouTube ReVanced for me and works fine without a license, they use the same type of "free but not really" that Total Commander or WinRAR use, sans nag screen. Add the FUTO F-Droid repository for easy installation and updating.
There's a bunch of other apps that I use that I didn't list since they're not as interesting, or they're cumbersome to deal with or they rely on your phone being rooted, and my setup is already unorthodox as-is.