There are hundred, if not thousand of computer-games out there. Although, only a few have a rare replayability, giving you weeks of enjoyment. Which ones? Here's my list:

1. Diablo II Lord of Destruction

This is one of my first, and greatest, game I've bought. Giving you five worlds to conquer, and three levels of toughness - this game is surely awesome.
The multiplayer part is not only free, but a real challenge. Since it's kind of old nowadays, you can get it for close to no cash at all. You should try it out!

2. Warcraft III Frozen Throne

This game is great because of it's mods. There are a huge number of custom maps out there, which does not only give you a new scenery - it also gives you new units. Everything is moddable. To be frank, I've never played the campaign. Never. Simply bought it to play WmW, DotA and so on.

3. Half-Life

This game's storyline is legendawesome! No other game have got me as hooked as this one. Not only will I buy any half-life game without hesitation - I'd love it as well.

Half-Life 1 is a FPS Sci-Fi Action game. Built on the GoldSrc game engine, it offers a huge ability for modification, some of which has become famous. CS, TFT, DOD and so on were all modifications in the begining. Don't forget to check out Zombie Panic, a really great mod out there!

4. Counter-Strike

I got to mention this one. I think i've played CS more than any other game so far, if CSS and CZ is included here. You got two teams, the terrorist (bad guys) and the counter-terrorists (good guys), and they shoot at each other. To make the players move their fat asses, an objective is added. But I think all you readers already knew that.
A great game. Perhaps that's why it's included in the e-sports? Yeah.

5. C&C Generals Zero Hour

Real-time Strategy, or RTS for short, is a fun genre. And a challenge for the mind. Not only is the units well balanced, it also offers the choice to use different playstyles.
Well, GLA (low-quality but cheap) tends to win most of the time, however it requires great skill and macro in order to use them properly.
I played a match against my friends not too long ago. Ugh. I don't like this game anymore, however, I used to love it a long time ago.