1. Half-life - Brilliant FPS, and one of the best games ever made. Re-defined the PC genre of FPS' and was light years ahead of it's time. A story that has been repeated a thousand times in shockingly bad movies and games. The sheer loneliness in Half-Life has never quite been rivaled since. Play it without the lights on, go on.
2. Doom - The grand-daddy of the FPS action genre. Nothing could quite compare to the demented nature of Id's difficulty levels, names such as "Hurt Me Plenty" will live long in the memory. Sadly the series never managed to surpass it's original and wonderful incarnation.
3. Marathon - Those of you who put Halo in your top five, well it's based on Bungie's 90's FPS: Marathon. Similar to Doom in graphics and style, it was a brilliant addition to the genre, and had a story that most films and books would be proud of. If you have never heard of the series, go play Marathon II: Durandal - amazing story.
4. Goldeneye - An FPS list wouldnt be complete without including Goldeneye, it's as simple as that. There arent many games which are so good, and so addictive, that it makes you want to just buy the entire system for that one product. GTA3 did it, you might argue that GTAIV will do it for the PS3, but Goldeneye certainly did it for the N64.
5. Quake - Excellent game that found its real home on the multiplayer platform. Days spent battling my mates on Quake II on an unreliable 56k connection years ago have long since imbedded Quake's undeniable impact on my brain. Of course, it's debatable whether the game should make it into a top five seeing as Doom is already in here, but meh.
Notable other - Deus Ex - excellent idea for a game, sadly its second incarnation failed to live up to the hype.