Oscar Hammerstein II.
If we might branch out a bit:
Three common, currently-circulating US coins bear two depictions of the same president. What are they?
(there used to be only one such coin, but they had to go and complicate a perfectly good trivia question)
The penny, the South Dakota quarter and the New Jersey quarter.
Who was the first American President? (This is a trick question.)
John Hanson. (Like nobody knows that. Sheesh!) 
Yeah, so I googled it. Sue me.
Robert Shaw and Richard Burton were both nominated for playing Henry VIII, but neither won (though Charles Laughton did).
And Iris Murdoch and Elizabeth I also fit this bill (as previously mentioned).
Janet Gaynor’s character won an Oscar in A Star is Born several years after Gaynor herself won an Oscar. In the remake, Judy Garland’s character won an Oscar, and Judy won a Special (Juvenile) Oscar.
3 - one competitive Oscar and two Special Achievement awards.
Who were the other 3 nobel laureates to get Oscar nominations?
Who was the only person to win an Oscar for playing a character named Oscar?
What are the only four movies to feature 3 of the 4 acting Oscar-winners from the same year?
Who was the only actress to be portrayed by another actress in a Best Picture winner before actually earning an Oscar nomination herself?
What was the first animated film to get a Best Picture nomination? A Best Screenplay nomination?
Billie Burke–played by Myrna Loy in The Great Ziegfeld in 1933 (34?)–nominated herself in 1939 for Merrily We Roll .
Depends on who you ask - Hanson was the first President of the United States in Congress Assembled after the Articles of Confederation were ratified by all states, but really, Maryland was the slowpoke - you could argue that Samuel Huntington was the first President under the fully ratified Articles.
I had to look it up too. It’s kind of sad that nobody remembers any of those guys.
ETA - it seems by Wikipedia that Hanson proclaimed the first national Thanksgiving holiday, called for the first national census, granted General Washington powers to negotiate for prisoner exchanges, established the US Mint, and did a lot of other useful stuff, which is pretty impressive seeing that those Articles Presidents didn’t get a lot of power.
A Streetcar Named Desire and Network are the two I’m sure of. Titanic was probably another one. And there’s some schlocky action picture made in the twilights of their careers that had three Oscar winners from the same year, but was not the movie that any of them won for.
I’ll post this, and then go look it up.
It’s really too simple, and I know everyone here will get it right off, but you’d be surprised by how many people I’ve stumped with this:
What is Scarlett O’Hara’s middle name?
Mudbone?
The four major actors of this classic Broadway play all eventually won Oscars, two for the film version, and two for other films. Interestingly, the one member of the cast who won a Tony Award for the role took the longest to win an Oscar.
Who was the first director to win an Oscar for the very first film he directed?
Scarlett, first name Katherine.
Scarlett, right? I haven’t seen the movie in ages, but her dad always called her “Katy Scarlett O’Hara.”
Scarlett is correct!
(But is “Katie” short for “Katherine?” Might it not be the Irish “Kathleen?” Does the book say for sure? I don’t remember.)
Two different long-running newspaper comics with the same title, one British and one American, debuted in their respective papers by absolute coincidence on the very same day. What is the title shared by these two comics?
Dennis the Menace. And they debuted one week apart, not the same day; the British verson first.
Name two families that each had three generations of Oscar winners.
Who were the only three women nominated for best director?