If you're looking at sports back to the founding, New York technically lost three pro teams before the turn of the 20th century. NL New York Mutuals were kicked out of the league, the New York Metropolitans folded when the American Association did, and the Players League New York Giants folded when the Players League did. Then the Giants and Dodgers evacuated New York, so that's 5, though the Yankees remained through that period.
But if you're looking at teams from the point at which the leagues stabilized, there are a number of cities that have had two teams fold or relocate, and get a third team back. In baseball, Milwaukee lost the Brewers and the Braves before the current Brewers, New York lost the Giants and Dodgers before getting the Mets as a second team, Washington lost two iterations of the Senators before the Nationals came in, there could be others.
In football, Los Angeles lost the Chargers to San Diego, then the Raiders and Rams, before regaining the Chargers and Rams, so there's 3 lost teams, Cleveland lost the Rams and Browns before the Browns "reactivated", Boston lost the Redskins and Yanks before the Patriots came into being, Baltimore lost two different teams named the Colts before they ended up with the Ravens, Dallas lost two different teams named the Texans, one of which became one of the aforementioned Baltimore Colts, the other is now the Kansas City Chiefs, there are others I'm sure.
In the NBA, which I don't know nearly as well, Chicago lost the Stags and the Zephyrs before the Bulls were created. San Diego lost two teams, but never gained back a third. Most other moves were one and done, but there are some franchises, like the Sacramento Kings, that have been itinerant as hell, and played in at least 3 or 4 cities.
NHL, I believe the answer is two, shared across three cities. Montreal lost the Wanderers (after 4 games!) and Maroons, though the Habs continued on, Quebec lost the Bulldogs and the Nordiques, and of course you know about Atlanta. Most other cases, starting from the foundation of the NHL proper, were single team losses later replaced: the Pittsburgh Pirates, Philadelphia Quakers, original Ottawa Senators, St. Louis Eagles, New York Americans, California Seals, Minnesota North Stars, original Winnipeg Jets, and Colorado Rockies were all replaced by new teams after the fact, though not as explicit replacements in most cases.
So, in short, if you take all of baseball history into account, it's probably NYC baseball; if you don't, it's probably Los Angeles football. Otherwise, you get a huge mess of "city lost two teams, now has a third."