I was about to ask why there is a giant penny and a dinosaur in the Batcave, but both of those questions have been answered already. The other question I had regarding the Batcave is this: I know that several villains, including Bane and Catwoman, have been inside the cave, but I don't remember ever reading that Batman's archenemy, the Joker, has also gotten into the Batcave. Has he?
TL;DR, Probably not in the comics
At the end of the "Death of the Family" arc, The Joker establishes an elaborate set-piece in the caves outside of the Bat Cave, and implies that he followed Batman back a long time ago.
In Batman #17, Joker hosts a mock dinner in the caves leading to the Batcave. Batman and his allies have been bound and doused in gasoline, with Joker warning against escaping at the risk of agitating the flint in the cave; everyone in the Bat-family save for Batman have also been bandaged. A brainwashed Alfred serves them with a cloche seemingly containing their own severed faces. When Joker threatens to ignite the gasoline, Batman escapes and triggers a blaze, but uses an explosive and his knowledge of the cave system to blow its roof open, allowing water to rush in from above and douse the flames. When he frees the others and they take their bandages off, it is revealed that they still have their faces — the whole scene was a sick ploy by Joker.
Batman pursues Joker to the edge of a large drop. Batman claims that he knows Joker's true identity, but to prevent him from saying it, Joker purposefully knocks himself over the edge and falls out of sight; Batman finds a notebook which Joker claimed held the secret identities of the Bat-family, but discovers it is blank. Meanwhile, the Bat-family are gassed into attacking each other, but manage to fight off the gas' influence. Later, in Wayne Manor, Batman tells a recovering Alfred that after finding the joker playing card in the cave, he confronted Joker about it in Arkham Aslyum as Bruce Wayne. Joker failed to acknowledge him and Batman realized that he did not care who Batman was under his mask.
Just to make it clear, the writer has since stated that Batman "has physical evidence that The Joker couldn't have been in the cave ... he's positive that this didn't happen".
As a short answer, there is at least once that the Joker has been inside the Bat Cave that I can recall. A semi recent cartoon movie, The Batman: Dracula, set in the The Batman cartoon universe, has the Joker, a ghoul almost full vampire convert of the titular and historical Vladimir the Impaler Dracula, locked up by Batman in a cell in the Bat Cave. He doesn't have the run of the place, but in it he is.