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2.
To bust one's balls is to mess or joke with someone.
Person 1: I slept with your girlfriend last night.
Person 2: Are you serious?! I'm going to kill you!!
Person 1: I'm just busting your balls! I would never sleep with your girlfriend.
by little t August 01, 2004
 
1.
"balls" is refering to someone's ego

so busting someone's balls would be making them insecure in some way

busting one's balls could be messing with them, annoying them, teasing them, embarrasing them, etc.

they use this phrase a lot in the movie "The Goodfellas"
by Josh June 23, 2006