1.VerbUsed as a polite formula to ask for a person's name, or to request that they join one in a dance, etc.
"May I have the pleasure? [ = an offer to dance]; May I have the pleasure of this dance?; May I have the pleasure of your company on the dance floor?; I'm sorry, but I don't believe I've had the pleasure. [ = I don't recall having met you or having been introduced to you before]; I'm sorry, but I don't believe I've had the pleasure of meeting you yet."