Mel Brooks. Writer, movie producer, director, actor, writer of hit Broadway musicals (including writing the songs)
John Lennon – songwriter and performer, actor (Including the non-Beatles * How I won the War*, and author of two critically acclaimed books (In His Own Write and A Spaniard in the Works).
Eric Idle - writer, actor, songwriter (for Spamalot).
Here’s the link that I meant to give in post #19:
Some great suggestions, and so many people I’d never heard of before to discover.
Looking over them there are a few actor-director-producer-writer combinations. Without discounting anyone’s achievements is this more of a phase-shift that happens in a successful long career compared to, say, Arnold Schwarzeneggar [body building - acting - politics] or my pin-up woman Geena Davis [acting - archery - social issues] that represent quite distinct domains?
William Blake: poet, painter, printmaker.
Frank Sinatra – singer, Oscar-winning actor, and painter/artist.
Canada Lee – Not well known today, but he had careers as a jockey, welterweight boxer, bandleader, actor (Broadway and movies), radio DJ (one of the first), and civil rights activist. His biography is fascinating reading.
John Lithgow is an actor, musician, singer, comedian, voice actor, and author. Pretty impressive.
Moby is, Wiki says, “a DJ, record producer, singer, songwriter, musician, photographer and animal rights activist.”
Brian Eno is a “musician, composer, record producer, singer, writer, and visual artist.”
Mikhail Baryshnikov is a “dancer, choreographer and actor.”
Glenn Close is a “actress, singer and film producer.”
He is indeed. We saw the Lawren Harris exhibit of which he was guest curator (mentioned in that article) at the Art Gallery of Ontario last year. Quite interesting, and they had his opening remarks on a video loop at the entrance.
Yep. He’s also a singer-songwriter, and at Howard Shore’s request wrote the melody that Aragorn sings at his coronation in The Return of the King (at 3:45 here; the words are Tolkien’s): https://www.youtube.com/watch?v=2jJnyyta07k
One guy who does: Peter Weller, who’s been teaching Art History for years. He’s also been directing for years – mostly television work, but also the project that got him an Oscar nomination for Best Short Film.
But the first line of his obituary will mention that he was the star of ROBOCOP.
Even if the actor-director-producer-writer combo deserves short shrift – and I’m not saying that it does – mention should still be made of actor-director-producer-writer Barbra Streisand, for doing award-winning movie work in multiple fields in her spare time from racking up an over-the-top level of success as, y’know, a singer.
Not completely distinct, but Jimmy Buffett, beyond his renown as a singer songwriter has written some novels, produced movie theme music, and acted in supporting roles in some movies.
He has also used the cachet of his song style to launch a chain of establishments, a line of food and beverages, and recently a retirement community for all those aging parrottheads. 
Although I suspect the advice of his uncle Warren was involved somewhere. :smack:
Gypsy Rose Lee – actress, novelist, playwright, talk show host. And, of course, a striptease artist.
Dolph Lundgren - World-Class competitor in Japanese Karate, 3rd Degree Black Belt; Masters in Chemical Engineering and awarded a Fulbright Scholarship to M.I.T; and actor/(drummer)/director/producer.
[sub]In Expendables 2 (I believe) the team leader remarks how [the character] has a Masters in Chemical Engineering and went to M.I.T.[/sub]
Also served as helicopter pilot in the US army.
Donald Glover. Standup Comic, Rapper, Singer, TV Show Writer/Producer/Director/Actor.
Besides Peter Weller, among the celebrities in artistic fields who have Ph.D.s (and not honorary ones) are Brian May (astrophysics), Dexter Holland (molecular biology), and Mayim Bialik (neuroscience, just like her character in The Big Bang Theory):
Natalie Portman is an actress, dancer, and contributed to two published scholarly journal articles. Fourth paragraph under “Education” here: Natalie Portman - Wikipedia
Not only that, but she is one of the few people to actually have an Erdős–Bacon number!
Her Erdős number is 5, and her Bacon number is 2, for a combined Erdős–Bacon number of 7.
Omer Sharif who is best known for his role in Lawrence of Arabia, for which he was nominated for an Oscar, spoke seven languages fluently and was a world class contract bridge player.
Author Jo Nesbø is also lead singer and guitarist for the band Di Derre, which has 4 top ten albums in his native Norway (the band’s first hit album predates the Harry Hole series.)
Bryan May, guitarist for Queen and currently holding the record for the longest time keeping the same hairstyle, has a PhD in astrophysics. Not the arts, but awesome in itself.