"There's No Business Like Show Business" is an Irving Berlin song, written for the 1946 musical Annie Get Your Gun and orchestrated by Ted Royal. The song, a slightly tongue-in-cheek salute to the glamour and excitement of a life in show business, is sung in the musical by members of Buffalo Bill's Wild West Show in an attempt to persuade Annie Oakley to join the production. It is reprised three times in the musical.

Based on this passage, who wrote the song there's no business like show business? Let's give stream of consciousness first:
The relevant information to answer the above question is: "There's No Business Like Show Business" is an Irving Berlin song, written for the 1946 musical Annie Get Your Gun and orchestrated by Ted Royal.
So, the final answer is Irving Berlin.