This is a two-fold request. First, list your favorite shooter already on GOG. Second, list the one shooter you would like to see brought to GOG.

My selections:
1. Medal of Honor: Allied Assault
2. Call of Duty (original, 2003)

GO!

Doom.

The only FPS I'll ever need.

The same answer to both:

Operation: Flashpoint GOTY

Though the Stalker games... yeah, pretty great.

1. Unreal Tournament 1 GOTY
2. SWAT 4 + expansion

Favorite on gog? Definitely Unreal

Can't really think of one I would want to be brought here though :P Well, Bioshock or Bioshock2 perhaps. Totally forgot!. Scratch Bioshock 1&2. I'd love to see Swat 4 here.

Stalker CoP, would be Arma Two *sigh*

Arma Three.

Exactly what I was going to say!

1. Either Blood or Shogo: Mobile Armor Division
2. Shadow Ops: Red Mercury

Well regarding the my must have list of Importan/Classic/Popular Old School PC games which I have categorized the ones that are FPSes:

Bohemia Library (Thus includes Arma games)
Half-Life 1 and 2
Counter Strike and Condition Zero
Doom 3 and Ressurection of Evil expansion
Battlefield 1942, Vietnam, 2, and 2142
Call of Duty 1+United Offensive
SWAT 4
Hexen series
Heretic Games
Marathon series
Tom Clancy's Rainbow Six 2 and 3 and their expansions
Brother's in Arms Road to Hill 30 and Earned in Blood
Tron 2.0
Red Orchestra 1
Crysis 1+Warhead
No One Lives Forever 1 and 2
Medal of Honor: Pacific Assault
Duke Nukem games

Out of all the ones there the one I want the most is:

(Drum roll)

Crysis 1+ the Warhead stand-alone expansion.

the original Doom (1993)

either Duke Nukem or Heretic

already here is Quake 3 Arena

And what I would love to see here is Quake Live or Unreal Tournament 3 Black Edition.

Doom/Doom 2 are the only FPS games you'd ever need.

But Gog needs Descent & Descent 2 again! Good thing I have physical copies of them.

Borderlands. For the realism, obviously. They're not on GOG but they should be. Tales from the Borderlands is, though. And that's quite good too, but not an FPS.

Already here : Serious Sam First/Second Encounter, Postal 2 or Sin gold.

Wishlisted : Duke Nukem Forever, Postal 2 Paradise Lost, Serious Sam 2 & 3, Doom 3.

My favorite shooter already on GOG is Strife.
The one I'd like to see here the most is Boiling Point: Road to Hell (aka Xenus).