A game series you need to play as a whole

Here's one for the surprise:
Post Mortem -> Still Life

Witcher games - some choices can change the storyline in the 2nd and 3rd game.

Mass Effect games - same as above.

Dead Space games - while those games don't have that storyline alteration thing, it's definitely a blast to play on hardcore difficulties for that deadly horror effect.

Orcs Must Die! games

Serious Sam games (for the first and second encounter games, you can choose either to play these in HD or not, but ignore the random encounter one...).

Half-Life games

F.E.A.R. games (Yes, all of those games. The 3rd one is disappointing, but still fun.)

Max Payne games

HITMAN games

That's pretty much I know what to recommend.

Honorable mentions: Crysis (1st was alright. Expansion as well. But from the 2nd game, you're going to be dramatically disappointed.)

Prince of Persia Trilogy. It's a masterpiece if it's a genere you enjoy.

As an avid Mega Man fan, the saga I always recommed is the Battle Network Series (6 GBA games), followed by the Star Force (3 NDS games) series. I always feel these are two very underappreciated games, and they are awesome for what they are, and its cool to see how they refine their mechanics each and each game. Most complains I hear is that the story is kinda "childish" and the card base mechanics are not "what Mega Man is all about", but for me was nice they tried something different, and the gameplay is quite enjoyable.

And if we keep going on Mega Man, you can always play the X saga, Zero saga and ZX saga in that order. They are all awesome games (of course, ones are worse than other coughX6coughX7*cough), and a (in)complete story as a whole.

And how am I going to forget my all time favorite Mega Man Games? The Legends Saga. Its a wonderfull 3D Mega Man Saga with charming characters, interesting story, cool gameplay and no conclusion whatsoever because this is a cruel unfair world.

As for the classic saga... I don't feel it has to be played from 1 to 6... or 8... or 10. Don't get me wrong, they are awesome games (the 3rd and 8th being my favorites), but compared to the other games in terms of following a story, you can jump from 1 to 4 to 6 to 10 and you wont miss much.

It seems like a lot of the ones I would have thought of have been mentioned. I'm going to throw out some random ones.

Jak and Daxter/Jak 2 & 3, Ratchet and Clank(ALL of them), Fallout(That includes 1, 2, tactics AND Brotherhood(For the sadistic ones) for you '2cool4oldstuff' people), Deus Ex.

All I can think of right now but these are ones that haven't been mentioned yet.

Fallout, all of them. Before one may also play Wasteland (The Classic), for all the references existing in Fallout.

Darksiders... I still got hope for other parts of the game.

Sid Meier's Civilization

Thief series

Thief: The Dark Project
Thief: The Metal Age
Thief: Deadly Shadows
Thief (2014)

The storyline will be very confusing unless you play the first game. You have the factions of Hammerites, Pagans, Keepers, and Mechanists (group that branched-off the Hammerites, leader is a mechanical genius named Karras).

The latest Thief game is a reboot of the original, but set much later than the earlier games. Unsurprisingly, most of the ideas are recycled from earlier games.

Resident Evil

Divine Divinity series
Drakensang with the 2. game played first
Final Fantasy 13/13-2
Overlord
Stalker series
Spellforce
Gothic
Risen
Quest for Glory
Dead Island

I've had a great time playing the Batman Arkham series (Didn't have the opportunity to play the last one but so far I've seen on youtube that it features a great story-line)

Monkey Island.

The Blackwell Trilogy. Pretty mediocre individually, but ties a broader story together quite nicely.

stalker

trine games are really good.

warhammer 40.000 dawn of war 2 series are good too.

tales of...... series are good (but only one in PC you should play whit emulator de ps2/ps1 games)

final fantasy series

bioshock games

If the title still applies, I'd like to recommend the Wizardry series. It's divided into 2 main segments - Wizardry one to 3 are classic dungeon crawlers with a little story, part 4 lets you take the part of the villain of the first part trying to escape the dungeon.
Now it starts to get really nice: part 5 to 8 tie together (as in the first games you can take over your characters) to form an epic arc. If you come to play it (and get the ending where the vampire princess hands you a ring in part 5) be sure to keep the char carrying it to part 8 ;)