In a nutshell
tinyRSS is a free, easily installed feedreader. It’s strengths aren’t its looks but it’s extensible design and power-user features.
What it does
TT-RSS provides a full-featured feedreading experience, including quite a few plugins, especially for sharing. It features a three pane layout that is similar to classic feedreaders like Bloglines or offline feedreaders of yore (anyone remember these locally installed ones?).
Compared to Fever or Selfoss it’s not exactly pretty, but it’s more powerful and adaptable to individual needs. It also features a mobile version, which may look a bit dated in its iOS design, but is very functional indeed.
TT-RSS features a plugin system, so certain things can just be added to tailor it more to your needs (I find it particularly great that they have a plugin to save links for later reading to OwnCloud, turning the whole experience into an even more self-hosted experience).
Even thought he good people who develop tinyRSS warn that it’s not exactly suitable for shared webspaces, it actually installed without a hitch on my shared webspace.
While it is rather simple to install, one should be comfortable with things like inserting SQL files into an existing database (if your webspace provider gives you access to phpMyAdmin, that shouldn’t pose a problem). Apart from that it’s a matter of uploading the files and editing a not-so extensive config file.
tinyRSS is free and open source.