What did the Joker want to prove in The Dark Knight?

Some obvious, important quotes:

• Introduce a little anarchy. Upset the established order, and everything becomes chaos. I'm an agent of chaos. Oh, and you know the thing about chaos? It's fair!

• Don't talk like one of them, you're not! Even if you'd like to be. To them, you're just a freak, like me. They need you right now. But when they don't, they'll cast you out, like a leper. See, their morals, their code... it's a bad joke. Dropped at the first sign of trouble. They're only as good as the world allows them to be. I'll show you, when the chips are down, these... these civilized people? They'll eat each other. See, I'm not a monster, I'm just ahead of the curve.

So, as Ionstar mentioned: the Joker wants chaos.

However, I don't agree that "he wanted to validate himself by dragging or proving everyone could be dragged down to his level". He needs no validations.

There is a quote which I think is more important than the two above:

Batman: Then why do you want to kill me?

The Joker: [giggling] Kill you?! I don't wanna to kill you! What would I do without you? Go back to ripping off mob dealers? No, no, NO! No. You... you... complete me.

Don't forget that the Joker is mad (at least when viewed form the perspective of us, civilized people). He wants to play with the Batman, probably because he sees him as the only worthy opponent (don't forget that the Joker effortlessly rips off whole mob of Gotham!).

Finally, the Joker believes that the he and the Batman are destined to do this forever (as they really do in the comic books). That's why he never kills the Batman (and it's not like he never had the chance): You won't kill me out of some misplaced sense of self-righteousness. And I won't kill you because you're just too much fun.

To conclude: it's just a challenge and good fun for him.

The Joker was a disciple of chaos, and wanted to prove that anyone - especially Batman, disciple of law - were as ready to abandon law and morality as he had already done.

1. He had the robbers progressively executing each other in the opening scene.
2. He forced Batman to choose between Harvey Dent and his girlfriend.
3. He went to Harvey in the hospital room and turned his anger from what the Joker had done to the arbitrariness of life, turning Harvey into Two-Face.
4. He threatened to blow up the hospital unless someone murdered the man who knew Batman's true identity.
5. He set up the boat bombs, to set people at each other's throats.

Basically, he wanted to validate himself by dragging or proving everyone could be dragged down to his level, and was pretty successful at it (even Batman, using the cell phone monitoring technology from Lucius). Even at the end, the Joker encouraged Batman to indulge his rage and drop him from the roof.

Or, as Alfred said, some people just want to watch the world burn.

What did the Joker want to prove? The long answer:

“You only have power over people as long as you don't take everything away from them. But when you've robbed a man of everything, he's no longer in your power--he's free again.” ? Aleksandr Solzhenitsyn

When people are attached to things (e.g. Batman to Rachel (and his rules), Harvey Dent to fairness, the mob to $$ and power), you have power over them with that thing. They can be held hostage. The Joker has no rules and nothing he's attached to, so he can't be held hostage for anything. He's free.

If he's trying to "prove" anything, it's to show how stuck people are based on the beliefs/attachments that they think will keep them safe and happy. He will prove them wrong every time. Even though he looks evil, he's helping people to become more free by taking everything away from them. (Another example of this is the imprisonment of Evie in "V for Vendetta").

"You can't take it with you." It's part of the Hero's Journey.