Which professional baseball team(s) have moved the most times?

The most any franchise has moved is two times. Following are the three teams that did so:

1. The Milwaukee Brewers moved to St. Louis and became the the Browns in 1902. Then the Browns moved to Baltimore and became the Orioles in 1954.

2. The Boston Braves moved to Milwaukee in 1953, then moved to Atlanta in 1966.

3. The Philadelphia Athletics moved to Kansas City in 1955, then moved to Oakland in 1968.

https://en.wikipedia.org/wiki/Relocation_of_professional_sports_teams#Major_League_Baseball

In Major League Baseball, the highest number of moves would be two (three cities involved), which applies to the current Atlanta Braves (Boston-> Milwaukee-> Atlanta), Baltimore Orioles (Milwaukee-> St. Louis-> Baltimore), and Oakland Athletics (Philadelphia-> Kansas City-> Oakland). See http://cjrtools.org/baseball/teams/history.html for details. If the Oakland Athletics do follow through on moving elsewhere (Fremont, Sacramento, or SanJosé have been proposed- https://en.wikipedia.org/wiki/Oakland_Athletics
they would lead this list.

Seven other teams have moved cities once (Dodgers, Giants, Senators/Twins, Senators/Rangers, Pilots/Brewers, Orioles/Yankees, Expos/Nationals).

These teams moved twice apiece (three different cities):

Milwaukee Brewers -> St. Louis Browns -> Baltimore Orioles

Boston Red Stockings/Doves/Rustlers/Braves -> Milwaukee Braves -> Atlanta Braves

Philadelphia Athletics -> Kansas City Athletics -> Oakland Athletics

https://en.wikipedia.org/wiki/Relocation_of_professional_sports_teams#Major_League_Baseball