I want to buy Crysis, but I see there is Crysis, Crysis Warhead and Crysis Max Edition. The thing here is that Crysys Warhead has a multiplayer component, but does it add anything extra to the story of Crysis, or does it only add multiplayer?
Crysis Warhead isn't a multiplayer component. It's a fully single-player game in fact, telling the story from the viewpoint of Psycho, with in my humble opinion better gameplay than the original.
You're talking about Crysis Wars, which is a separate game bundled with Warhead, as its multiplayer component. This is in contrast with Crysis where the multiplayer was integrated.
I would say getting Warhead is definitely worth it to add to the story and have a multiplayer game.