Unfortunately Replicant does not run on my phone. When time comes to purchase a new one, I will probably get a compatible one. Most of the apps I use already come from the F-Droid repository https://f-droid.org/repository/browse/ as there is a lot of fantastic software available there.
I don't like it when threads end up as list of lists but some recommendations that might not be well known:
Aard is a offline wiki(pedia and others) reader. AFWall+ is a firewall to give/revoke internet access to apps. It is a fork of DroidWall which went inactive. Document Viewer is the best PDF reader I know and it's free software, hooray. It can automatically cut off white margins, you can set your own touch controls etc. Gibberbot is a great XMPP client. I tested many but this one did not randomly stop receiving messages like all the others did. GnuCash for keeping track of finances. Mirakel is a Todo app which I switched from Astrid to. Astrid is free software too but relies on several non-free libraries from what I know. Mirakel seems more concerned with privacy. OsmAnd for offline maps. And of course you can get APG, AdAway, ConnectBot, FBReader, K-9 Mail, ScummVM, Terminal Emulator etc. from F-Droid.
Really BAD: has too much proprietary things and the hardware is bad for freedom: modem transport is over HSIC (high speed USB), but it controls the GPS (included in the modem chip), audio and perhaps NAND too
