The Joker is one of Batman's greatest enemies and one of the evilest villains in comic books, but he's rarely given a concrete reason for the things he puts Batman through. Joker did eventually admit to what he truly wants to Batman though, and it's the most perfect reason he could have.
Batman and Joker's history has been long and brutal, they've been fighting each other in comics for almost 80 years at this point. However, during this time, Joker's real reason for wanting to torture Batman wasn't really elaborated on all that much. Some people think Joker wants to kill Batman and destroy Gotham, some people think Joker wants Batman to kill him in order to break Batman's one rule. But the actual truth of what the Joker wants is just a bit more complicated. Despite Joker's focus on killing Robin, he doesn't actually want to push Batman to the limit, he wants the exact opposite.
After several decades of fighting against the Joker, Batman finally asks him what he actually wants from him in The Batman Who Laughs #4 by Scott Snyder and Jock. Joker's answer is surprising in how honest it is, Batman questions him, "You want me to win?" Then, Joker responds in complete honesty, possibly for the first time in his life, "I want neither of us to win, ever." Joker makes it clear that the only thing that gives Joker's life meaning is his battle against Batman. But, the idea that Joker wants to fight Batman forever isn't new. In Christopher Nolan's The Dark Knight, Joker openly says "I think you and I are destined to do this forever." While the idea itself isn't necessarily new, this is one of the first times in comics that Joker has openly admitted his hopes to Batman.
How Batman Reacts to Joker's Creepy Request
Batman seems to take the news actually rather well as he helps Joker stand up and refers to him as a friend (but his politeness could be a result of the massive amounts of Joker toxin currently pumping through Bruce's veins). Joker admits that all he's ever wanted was to stay near Batman and continue interacting with him. In fact, Scott Snyder's entire run on Batman focused heavily on the idea of Batman and Joker being engaged in a twisted version of friendship.
Why Joker Can't Exist Without Batman
The one time Joker did feel like he succeeded in killing Batman, he immediately hung up his clown makeup and tried to create a new life for himself without the Joker identity as seen in Batman: Going Sane by J.M. DeMatteis and Joe Staton. It has been proven that without Batman Joker can not exist. Who would Joker be without his nemesis? How would he spend his days if he wasn't busy trying to figure out how to antagonize Batman? Joker relishes in his ongoing fight against Batman because it gives his life purpose, which is why Joker wants his battle with Batman to never end.