Ok lol it's not easy to answer.
First off let me state these incontrivertible facts:
Q3 > Q1 > Q2 > UT
Now;
Between 2012- and 2016 the best fps was BLR, but sadly that game is no more.
QuakeLive or QC are the best classic fps. Reflex Arena is the best arena shooter which should mean the same as "classic fps" but Reflex has CPMA movement while the Quake franchise has instead progressed in dumbing down the movement - which is what was happening with classic fps anyway.
Then you got a whooooooole lot of *new* fps, where it's a first person shooter, but other mechanics outside of move/aim/shoot are a much bigger factor.
There's tactical shooters, where the best is probably Insurgency, mil-sim where ARMA 3 is the king, there's DOTA style shooters with Overwatch and Paladins and a million more, casual shooters with TF and Dirty Bomb and Loadout and even PvZ ..
What matters to make a distinction is what YOU think a perfect fps needs; how low of a ttk, how difficult, how much randomness, how much health matters, how much speed matters, teammates, duel, kill range, pickups, map control, positioning, and so on.
Also consider how good the non-game elements are, such as community or matchmaking.