Good idea of @haknor to create this topic which should be perhaps linked to the “Using/e/” section so it gets more visibility. Certainly new users need recommendations like this list if they are not already familiar with this OS or FOSS apps.
I think all the apps I mention here are open source. One or two like OsmAnd~ have reservations marked on F-droid about them, but nothing that I consider serious breaches of security IMHO.
Worth mentioning perhaps for those wishing more advice about the privacy aspect of apps: G-Droid
It is based on F-droid and uses, I believe, the same repository, but it adds more detailed recommendations, with some user comments as well as a star metrics rating system to indicate how safe an app is, if it is updated regularly, etc. Its methods and classification system (e.g. suggestions for similar apps to those already installed) could be an inspiration for the App manager on /e/
Apps I use regularly:
For increased security,
ClassyShark3xodus (excellent for testing apps for leaks)
AFWall+ (needs rooted phone - I use Magisk for this)
For normal use,
Browsers: Fennec F-Droid, Firefox Klar (the latter for one-off use as it has no bookmarks)
File Manager: Ghost Commander (excellent SFTP plugin for transfer to another server)
Mail App: K-9
App store: F-Droid, G-Droid, Aurora Droid, Aurora (when it works) also see below
Keyboard: AnysoftKeyboard (great plug-ins for changing languages)
Photo Editor: Simple Gallery Pro
Note App: Joplin (excellent tool for cloud synchronisation with encryption)
Text editor: Markor (also useful as it creates files with markup language reusable with Joplin)
Ebook reader app: Book Reader
RSS reader: Feeder
Cloud synchronisation: Nextcloud
System info: CPU info
Password manager: KeePassDX
App installer: SAI
Another specialised App store for more security-minded geeky users: Nethunter Store
Many useful tools for testing your LAN & WiFi security and reading system logs. Hash Droid for verifying checksums on downloaded files; the latest Orbot for those who want to use Tor browser proxy system & many more security checking tools, many from the Kali Linux OS.
Hope this list (as well as others listed under this topic) may be useful to new /e/ users.