Meaning, all four of the major entertainment awards: Oscar, Emmy, Tony, Grammy.
I know Liza Minnelli has done it (for Cabaret, Liza With A Z, Flora the Red Menace, and a special Legendary Performer award, respectively). Who else?
(Bit of trivia: Liza is also the only person to have won an Oscar both of whose parents have also won Oscars (Judy Garland won an honorary Academy Award for Wizard and Vincente Minnelli for directing Gigi)
Rita Moreno. I know her Oscar was for West Side Story but I’m not sure what she won the other awards for.
As mentioned in the Comic Genius thread, Mel Brooks has won all four. The success of the stage version of The Producers put him over the top, I believe.
Mel Brooks
John Gielgud
Marvin Hamlisch
Helen Hayes
Audrey Hepburn
Rita Moreno
Mike Nichols
Richard Rodgers
Johnathan Tunick
If you’re counting special awards, Liza and Barbara Streisand both qualify. If you count daytime Emmys, Whoopi Goldberg is in too.
According to IBDB.com and IMDB.com, a total of 3 Tonys (all for The Producers), 1 Oscar (the original The Producers movie in 1969), 3 Emmys (all for guest spots on Mad About You), and 1 Grammy* (again, for the stage version of The Producers).
(Since IMDB is notoriously incomplete, it should be mentioned that he won a second Grammy in 1998 or 1999 (depending on the website) for some recording or another of The 2000 Year Old Man)
(But if we want to include Golden Globes to make it a round Five Awards, Mel has been nominated four times but never won)