One of the Joker's one biggest claims to fame is the fact he murdered the second Robin, Jason Todd, but looking back at the event shows Joker never actually wanted credit for his most famous crime at all.
The Joker has done an unbelievable number of evil things, from trying to poison the Gotham City water supply, stealing Bruce Wayne's fortune, taking over the city of Gotham and even killing Robin. Usually when Joker commits a crime it's because he wants Batman specifically to know it was him. Joker's entire deal is trying to push Batman to the edge. While Joker wants to fight Batman forever, he isn't above making Batman's life miserable during that time. The biggest thing Joker ever did that hurt Batman far more than anything else, was the killing of Jason Todd. Joker killing Jason was an iconic moment in comics and further developed Joker's relationship with Batman, but it turns out that originally, Joker never wanted credit for it at all.
While Joker killing Jason is an iconic moment now, it wasn't Jokers original intention for him to be found out by Batman. Joker entirely intended for Jason to die and for Batman to never know how or why. Despite the fact Joker could be redeemed according to DC, he's still the darkest and most evil villain in Batman's rogues' gallery and this is shown fully with his original plan for Jason. Joker's original idea was revealed in Jim Starlin and Jim Aparo's legendary Batman: A Death in the Family. After beating Robin half to death with a crowbar, Joker tries to use a bomb to not only kill Jason, but to also wipe out any and all evidence of Joker ever having been there. This is a huge element of the story that's often overlooked, because all subsequent depictions of the event frame it as if Joker killed Jason to intentionally send a personal message to Batman, but this shows that Joker never intended to take responsibility in the first place.
Joker Wanted The Mystery Of Robin's Death To Haunt Batman
DC has changed events now to lean more into the idea that Joker always wanted to kill a sidekick, recently Joker has tried to kill Boy Thunder, Superman's current sidekick, in the exact same manner that he killed Jason Todd. It's likely that originally, Jim Starling had Joker hide his involvement with Jason's death, so Batman could flex his detective skills and discover who was behind it, which he does over the course of Death in the Family. But DC seems to have realized that it's a much better and darker character moment if Joker not only kills Jason but is brutally upfront and honest about being the one to do it. Later DC Comics even have Joker attacking other supervillains if they kill a Robin, something Joker claims is only his gimmick to do.
Originally, Joker wanted absolutely no responsibility for Jason Todd's death, but overtime DC has leaned more into it and allowed Joker to truly claim responsibility for what became one of the absolute worst days in both Batman and Robin's life.