Well i can't give you the best FPS game ever because we will all disagree on that, but i can give FPS games that are really exceptional in my opinion.
Timesplitters, like a few said above it didn't have a serious story, gameplay was excellent, so it was a fun shooter which i really miss.
Killzone series, i really think this series can do better as mentioned here: http://www.vg247.com/forum/topic.php?id=9361
But for me the highlight of the series is the soundtrack by Joris De Man is phenomenal and epic, the serious story, the helghast opponents, the long ongoing war between ISA (In Shadow Fall its VSA) and Helghast, the character and background history, so all in all i think Killzone is my go to FPS game.
Medal Of Honor (Before it went crap): MOH 2010 was a good game and enjoyed playing it at the time, theres not much i can say it was only two games in the series i have played.
Red Faction: I enjoyed playing it back on my PS2 and i also enjoyed RF Guerrilla, that story was amazing to go through and the building were amazing to go through.
Resistance: Now i feel like i never gave this series a chance i have played Resistance Burning Skies plus Resistance 3 and didn't have that feeling like when i was playing the other games, really need to try it again though.
Far Cry: I Have only played FC3 and i have to say it is amazing although i think its mainly to do with Vaas, i enjoyed the dual endings which really reflect what kind of character you are at the end, the soundtrack is epic and emotional at the right times and i plan to try out FC3 Blood Dragon see how it is.
Wow, i play a lot more FPS games than expected.