Green’s Dictionary of Slang

1. (US) a superior person or one who claims to be so.

1948		Lait & Mortimer N.Y.: Confidential 121: [Street gang] war counsellors [are] known as ‘bigs’.
1955		B. Schulberg On the Waterfront (1964) 17: At the top of the heap the real bigs like Tom McGovern.
1972		N.Y. Daily News 20 July 14: Mob Big Denies Tie to Sinatra [HDAS].
1983		National Lampoon Mar. 57: Lydia [...] is the ten-year-old daughter of a Politburo big [HDAS].
1988		Forbes 14 Nov. 16: Long before Joyce Jillson became famous as a White House astrologer, she had been telling Hollywood bigs what their stars held for them [HDAS].

2. (US) in any important environment, in baseball, the major league .

1985		F. Deford in Sports Illus. June 🌐 The book on Greenfield Jimmy Smith as a ballplayer was good mouth, no hit [...] His major talent earned him another nickname up in the bigs, Serpent Tongue.
2022		A. Kirzman quoting Houston Chron. 30 Mar. in Giuliani 210: Oxford said he was looking forward to Giuliani showing his firm ‘how it is to play in the bigs’.

3. (US) in pl. constr. with the, an important company or organization.

2018

‘Aidan Truhen’ Price You Pay 100: But you got this itsy bitsy practice. I can’t work for the bigs I got kicked out.