This question originally appeared on Quora. 
Answer by Jesse Richards, Author of The Secret Peace
Because the Joker wins if Batman kills him. That's what the Joker wants. Everything he does is to taunt Batman into killing him. In fact, the interesting part of their relationship, the real conflict of each story, is not to see if Batman will stop him (he will), but to watch Batman struggle with not killing him, because anyone other than Batman would of course kill him. This self-control is Batman's superpower.
The Joker and Batman are each trying to prove a point to society - and really to us, the readers. The Joker wants Batman to kill him because he perfectly embodies chaos and anarchy and wants to prove a point to everyone that people are basically more chaotic than orderly. This is why he is so scary: we are worried he may be right. If the Joker is right, then civilization is a ruse and we are all truly monsters inside. If the Joker can prove that Batman - the most orderly and logical and self-controlled of all of us - is a monster inside, then we are all monsters inside, and that is terrifying. The Joker is terrifying because we fear that we are like him deep down - that he is us. Batman is what we (any average person) could be at our absolute best, and the Joker is what we could be at our absolute worst. The Joker's claim is that we are all terrible deep down, and it is only the law and our misplaced sense of justice that keeps us in line. Since Batman isn't confined by the law, he is a perfect test case to try to get him to "break." The Joker wants Batman to kill a person, any person, but knows that the only person Batman might ever even remotely consider killing would have to be a terrible monster, so is willing to do this himself and sacrifice himself to prove this macabre point. Batman needs to prove that it is not just laws that keep us in line, but basic human decency and our natural instinct NOT to kill. If Batman can prove this, then others will be inspired by his example (the citizens of Gotham, but again, also the readers), just as we are all inspired every day to keep civilization running smoothly and not descend into violence, anarchy, and chaos. This ability to be decent in the face of the horrors and temptations present all around us is humanity's superpower, the superpower of each of us. The struggle of Batman and the Joker is the internal struggle of each of us. But we are inspired by Batman's example, not the Joker's, because Batman always wins the argument, because he has not killed the Joker.
This basic logic applies to all superheroes who don't kill, but the Joker-Batman conflict is the most perfectly distilled example. There are a lot of other good answers on this page, and they are all different-but-correct ways of looking at the question, but to me, the philosophical and thematic reasons above are more resonant than the plot and character reasons that exist within the logic of the story.
Recommended reading: The Killing Joke (1988), obviously, but also Legends of the Dark Knight Annual #1 (1991, which the above picture is from, by Denny O'Neil and Jim Aparo), if you can find it. Excellent. And of course, also watching The Dark Knight. Nolan and Ledger got the Joker perfectly right.
Perhaps this is why sidekicks work so well with Batman. It's always been a bit of a conundrum why the "lone avenger" ended up with way more sidekicks and assistants than any other superhero. But if Batman is all about inspiring others, these partners support his cause as evidence that he is winning. (Being inspired by Batman is even part of the first Batgirl's origin.) Robin, in particular, needs to be young to show that Batman is inspiring the next generation. The Tim Drake Robin (my favorite) also shows that you can be inspired to help even without Batman's defining tragedy in your life. Perhaps this is why the ending of Miller's The Dark Knight Returns is so great, and why Morrison created Batman Inc., showing Batman inspiring others throughout the world.