For this list I will be focusing primarily on Open Source apps available through the F-Droid distribution platform. In no particular order of one over the other let's have a look!
This one is mostly relevant for people using a Keepass file to securely store and access their passwords and secrets.
This podcast app is quite amazing. It sports all the features you expect from a Podcast player/manager plus a fair bit of extras. It's quite feature clean in that it doesn't bother with useless gimmicks and focuses strongly on the main use-case; listening to podcasts and managing your subscriptions. Easily one of my favorite apps.
The official client for accessing your Nextcloud instance on Android.
This app is perfect for keeping directories in sync, something that is missing from Nextcloud. Syncthing and Nextcloud can be used complimentary.
5) Fennec (Firefox)
When it comes to privacy features in browsers it's hard to beat Firefox. However Firefox official builds do come with some proprietary blobs from Mozilla. If you are a Open Source purist you might want to go with Fennec instead. It is built from the Open Source Firefox codebase but stripped of it's proprietary blobs.
6) OpenVPN for Android
One of the core principles of security is trying to eliminate trust. When it comes to VPN's however trust is actually often still required. By using a 3rd party VPN you inherently trust the VPN provider to not keep logs like they claim and to not mess with the payloads. However "trusting" your VPN client is probably just as important. This is where OpenVPN for Android comes in.
Being Open Source and peer reviewed means you don't neccesarily have to trust in the developers claims so much as you can inspect it's source code to verify. One of the reasons I recommend installing these apps through F-Droid instead of Google Play is the fact that F-Droid only serves reproducable builds. This means all apps are/can be built from their respective source code in the repository. Apps can be Open Source but it's sometimes hard to verify a binary was actually derrived/built from that published source.
This app syncs with your wearables and exposes the fitness data from within the app itself instead of relying on some cloud service. If you own one or more of the supported hardware devices (Mi band for instance) this app allows you to do away with the proprietary apps that these devices normally sync with. Your health data oftenly gets sucked into some cloud service and is being aggregated and stored in purpetuity. If you care about your privacy but still wish to use wearables to track your health then this app is perfect for you.
Despite the slightly weird name this app is actually essential for privacy aware Android users that wish to sync their contact data through a CalDav server (e.g. Nextcloud Contacts). This is the perfect alternative for something like Google Contacts.
9) Aurora Store / Aurora Droid
When running a de-Googled/AOSP version of Android by default you have no access to Google Play app distribution. This is where Aurora Store comes in. Aurora let's you either use a generated anonymous Google account or let's you log-in to your own Google account and grants you access to all goodies the Play Store has to offer.
Ever wanted to control which app can access the internet or your local network. With AFWall+ Firewall you can easily block or permit access to several network interfaces on a per-app basis. You either use whitelisting or blacklisting strategies to gain fine-grained control over network access levels. This is especially useful for blocking some app that's not even supposed to be using LAN interface from accessing your local network. You would be surprised how manny apps actually wish to aquire said access.
11) Scrambled Exif
This app is an absolute gem when it comes to it's implementation. Have you ever considered the fact that with every photo you take on your device it actually attaches a LOT of information about you and your device. This metadata is called EXIF data and it often contains (depending on settings) your device name, type, location and a bunch more relevant data. This is mostly desireable for helping to archive and sort photo's as well as getting the right orientation (portrait/landscape) etc. However every time when you upload a photo to some service (say, WhatsApp) you rely on either the decency or the quality of said service to strip-out and/or not expose this information to a third-party.
Actively developed OTP authenticator. Even though this is a crowded space this landscape is riddled with apps that either don't use encryption to store your token secrets or received their last update several years back. AndOTP is very much alive and high in quality.
The accompanying app to the Wallabag self-hosted alternative to Pocket/Instapaper.
14) Element (Previously Riot)
Messaging client for the Matrix protocol.
This is the golden standard for YouTube apps on a vanilla Android system.
16) Slide for Reddit
Beautiful and feature complete Reddit client.
Beautiful and feature complete Twitter client.
This is the end-all-be-all of terminal emulators for Android. It sports a package manager and has a nice ecosystem of plugins surrounding it.
Even a PiHole is not perfect and occasionally ads will find their way to your device. This is where AdAway comes in as a knight in shiny red armour to save you from those pesky ads.
20) Loyalty Card Keychain
If you enjoy discounts but hate to carry around a wallet full of loyalty cards than this app is for you. This app makes it easy to scan (or manually add) the bar- or qrcodes from you physical cards and store them neatly and securely into one app. Believe me, it saves a lot of space in your otherwise cluttered wallet.