As one of Gotham's most feared crime bosses, there aren't many villains who The Joker may respect, but DC has confirmed that he respects at least one: The Riddler.
Joker and Riddler are frequent foes of Batman who have been tangled in each other's hairs as frequently as they have The Dark Knight's. While they often find themselves on the same side, the two have butted heads on numerous occasions. The most notable would be during the "War of Jokes and Riddles" arc during Tom King's Batman run. While the two criminals formed an alliance against The Caped Crusader during his second year on duty, The Joker and The Riddler's disagreement on who should kill Batman ultimately leads to a gang war between their two crews. This arc showcased how two egomaniacs can't bring themselves to put their egos aside in order to co-exist for a common goal. Yet, one still manages to respect the other.
In Batman (2011) #15 from Scott Snyder, Greg Capullo, Jonathan Glapion, and James Tynion IV (all a part of the Death of the Family arc), there's a story near the issue's end titled, "Red Light, Green Light." In it, The Joker breaks The Riddler out of Arkham Asylum, encouraging him to leave his cell before a Joker Bomb kills him. The reason why it's so important for The Riddler to him is that The Joker believes he sees the bigger picture of tormenting Batman better than most people do.
The Joker and The Riddler are two very similar sides to the same coin and Joker recognizes that. At their core, both characters are sociopaths with obsessions for Batman that go beyond just wanting to kill or defeat him. These villains would rather challenge The Dark Knight. The Joker challenges Batman's moral code while The Riddler challenges his wits. In The Riddler's case, he's smart enough to outsmart Batman and get away with his crimes if he didn't leave clues to them. But then again, there's no fun in his crimes without an audience to solve them, right? In the same way that The Joker takes the enjoyment out of trying his best to get Batman to break his code, The Riddler enjoys trying his best to stump The Bat.
What's impressive about this to The Joker is that out of Batman's entire rogues' gallery, Riddler is the only one smart enough to challenge Batman on a truly psychological level. Even better, Joker believes The Riddler's riddles make Batman smarter and dangerous, making The Joker's challenges all the more fun. This story recognizes that both characters, even when they can't get along, have similar goals in trying to mentally break Batman and how their different methods contribute to inadvertently helping each other achieve that goal. It's no wonder why The Joker would have such profound respect for The Riddler when he feels as though he sharpens his favorite Bat.
