Once. You swing and miss each time you serve, and your opponent commits a double fault each time he serves. Eventually you find yourself serving at 6-5 or 7-6 in the tiebreak, and then you heroically serve an ace.
From Dick Hess, All-Star Mathlete Puzzles, 2009.
UPDATE: A reader points out that a superlatively lazy player can win a set without ever hitting the ball, thanks to the hindrance rule:
If a player is hindered in playing the point by a deliberate act of the opponent(s), the player shall win the point.
“On the crucial tie-break point, your opponent dashes across the net and tackles you, awarding you the point and the set.”