My favorite FPS is probably battlefield 2142. Except this one, I've played descent (I and II), doom2, duke nukem 3D, half life, L4D, UT98, counter strike, smokin guns, unvanquished when it was close to tremulous, current unvanquished, red eclipse I and II, and others I forgot the name.
In the FOSS area, I think my favorite FPS is red eclipse I, for it's interesting movement mechanism. Also, the physics is rather good, despite the fact no physics book was mistreated while developing it (according to a comment seen in code).
I like unvanquished, despite it's gameplay flaws, because it's currently the best FOSS FPS game still developed (AFAIK) which brings in:
- 2 different gameplays (despite the balance problems this brings)
- a rather realistic movement mechanism (despite it being still quake with the lack of physics engine behind)
- real team play
- the RTS elements are not unpleasing neither, at least on paper (still lot of room to improvements on that)
It also provides server-side mods (like all quake-based games), which makes it easier to do experiments while allowing the use of a rather modern language for that (C++14 max, for now, maybe soon more recent versions will be possible, with the planned move to WASM). The kind of stuff I'd love to hear other, more recent engines allow.