Gosh, if this one ain't embarrassing...
So, as it turns out, while somewhere in that troubled mind of his, the Joker knows that Batman is Bruce Wayne, and that therefore Bruce Wayne should be considered an enemy, or at least mildly obstructive, the Clown Prince actually considers Wayne to be his friend.
No, really! There's very little research possible to explain a mind like the Joker's, but it's true. While Batman, and certainly therefore Bruce Wayne, considers Joker to be a homicidal menace, Joker himself actually views Wayne as something of an asset. You see, while Wayne has carried on the family legacy of funding the future of Gotham, there's any number of ways that the Joker can think of where Wayne is actually helping him most of all.
The most obvious way is that Wayne is a distraction, time spent away from the cowl and immediate business of the Dark Knight. The Joker doesn't do distractions. He doesn't even admit setbacks. He's constantly recalculating. There's a reason why he's Arkham's most escaped inmate. A man without real plans and without a real future, much less a real past, that's a man who is truly unhinged from just about every possible iteration of a normal life. No amount of psychological study can account for the Joker. He's not even a sociopath. He can't be categorized as crazy. He operates on a completely different playing field. If you asked him nicely, he might even admit that Batman himself is a friend.
But we're not here to discuss that. Right now, the subject is Bruce Wayne, and Bruce Wayne alone.
So, for a man who doesn't acknowledge distractions in his own life, the living personification of the distraction in what would conventionally considered his mortal enemy probably wouldn't be the explanation to go with. Wayne is not the Joker's pal because of or despite the connection with Batman. Perhaps he values the idea of the split personality, but again, the Joker doesn't display any regular pattern for that, either. He makes that up as he goes along.
It's worse than that. It's not even to do with all his wealth.
It's the fact that Wayne is an almost completely isolated man. To the Joker's mind, Wayne does everything to make sure everyone leaves him alone. If there's any method to the Joker's madness, that may be it, the explanation, the hole in things, the missing piece. In order to get what he wants, Wayne is willing to go to extraordinary lengths, and so is the Joker. Again, not necessarily to do with Batman or the privilege of wealth. Very few people dedicate themselves so thoroughly, regardless of the means or ends, as Joker has found in Bruce Wayne.
You might find a picture of Wayne on the Joker's nightstand, if he had one. It's doubtful that he even sleeps. If there's any mask that the Joker puts on, it's probably to attend the latest Wayne press conference, hiding in plain sight, another fine joke. In fact, the Joker, when he isn't locked away, should always be the most visible man in the world, and yet, somehow he isn't. He doesn't lack in subtlety, instead when it amuses him. And Bruce Wayne amuses the Joker. He's the Clown Prince's pal, if he needed one.