This one's actually a little tricky. No one had won three Oscars for a single film when Leo McCarey won Best Director and Best Original Story for Going My Way in 1944, a film he also produced. Going My Way was also named Best Picture, and since that award usually goes to the producer, McCarey would seem to be the answer.

Not quite. In the days of the studio system, studio heads sometimes accepted the Best Picture Oscar instead of the producer. This was the case with Going My Way, when Paramount head of production Buddy DeSylva accepted the award instead of McCarey (Bob Hope was on suspension by Paramount at the time, and when DeSylva went to the stage to receive the Oscar, Hope got on his knees and feigned shining the executive's shoes).

Billy Wilder was the first to actually take home three Oscars for a single film, for writing, directing and producing the Best Picture of 1960, The Apartment; and he is cited by most sources as the answer to the question. But since we made you read this lengthy answer, full credit if you said McCarey instead.

 
The first person to win four Oscars for a single film was Korean filmmaker Bong Joon Ho, who took home Best Picture, Best Director, Best Adapted Screenplay and Best International Feature for Parasite.