The Joker is well-known by comic book fans as Batman's insane enemy...but what specific mental illness does he actually possess? The Clown Prince of Crime has seemingly broken ever conceivable law in Gotham City, ranging from petty theft to mass murder on a metropolis-wide scale. All of his crimes are linked through his twisted sense of humor and his desire to play "jokes" on his victims - and therein lies a clue to the Joker's specific mental health illness (and whether or not he can actually be cured).
Diagnosing the Joker can be difficult because the character has changed drastically over time. His first appearance in Batman #1 depicted a criminal that showed little non-neural normal qualities. During the 50s and 60s, the Joker's crimes were limited to glorified pranks thanks to the draconian Comics Code Authority - and thanks to a rule stating that a criminal in a comic book must never get away with a crime, the Joker was arrested at the end of every issue (this cemented Arkham Asylum as an easily-escapable prison in the eyes of fans, perhaps damaging the CCA indirectly with every Joker story sold).
Bipolar Disorder is a common diagnosis among fans; the Joker can indeed be possessed by sudden mood swings and anger - but that doesn't explain his propensity to murder (or show little to no remorse after the fact). Dissociative Identity Disorder (incorrectly described as Multiple Personality Disorder) is often brought up as well, but the Joker has rarely shown other personalities like the split that defines Harvey Dent and Two-Face. But one medical diagnosis describes the Joker's symptoms to a T: Witzelsucht.
Those who suffer from Witzelsucht (German from "addicted to jokes" or "joke addiction") feel compelled to make puns and tell inappropriate jokes at inappropriate times. They often have frontal lobe damage and an altered sense of humor; they find slapstick and puns funnier than other, long-form jokes because they often have trouble connecting the setup of a joke to a punchline. This means they are perfectly capable of telling a joke, but have difficulty understanding the jokes of others. Joker rarely laughs at the jokes of others (unless they're the butt of a joke thanks to their own actions) and pain is indeed a large part of slapstick comedy. Unfortunately, it appears the Joker can't tell how much pain is too much - and even murder is funny to him.
There is a treatment for Witzelsucht: the antidepressant Venlafaxine can alleviate symptoms in some patients. Unfortunately, the diagnosis is incredibly rare, and a small sample size makes extensive studies difficult. If he does indeed suffer from Witzelsucht, there is still hope for the Joker - but his case is particularly severe, not to mention dangerous for anyone attempting to help him.