I realized that I think of Jack Nicholson by far foremost for his portrayal of (clinically or comically) insane characters.
What movies has he played such a character, and (roughly) what proportion of his roles do such characters compromise?
First, a bit about insanity. Currently insanity is not a term used by the medical profession. It is used by people in general to refer to mental illnesses that are no longer grouped as a single condition by medical professionals. There is also a legal definition of insanity.
On IMDB I count 65 feature films or made for TV movies with Jack Nicholson in them. He did not star in all of those. As far as I can tell, when he did not star, he was never portrayed in as mentally ill.
As far as I can tell, Jack Nicholson has one starring role where his character is clearly intended to be mentally ill. Of course that is:
Beyond that, there are two starring roles where his character starts the movie mentally healthy (as far as the viewer can tell) and is driven to what most people might call "criminal insanity" by other forces:
And finally, he masterfully played the role (which might have started the whole "insanity" casting of Jack Nicholson) of a man pretending to be insane (in the legal sense) to get out of serving jail time:
The proportion of actually mentally ill characters then is 1/65 or about 1.5%. For characters that exhibit authentic behavior that a reasonable person would describe as "insane", we can say 3/65 or about 4.6%. And if we want to just look at Nicholson's ability to act like an insane person (or even more challenging, act like a person acting like an insane person), we have 4/65 or 6.2%.