The 1916 Giants won 26 games without a defeat, BUT they did have a tie. The 1935 Cubs and the 1880 White Stocking had 21 in a row.
Here are the teams who have won twenty or more consecutive games from either league:
Team - Wins - Ties - At Home Wins 1916 New York Giants 26-1-26 1935 Chicago Cubs 21-0-18 1880 Chicago White Stockings 21-0-11 1884 St. Louis Maroons 20-0-16 1884 Providence Grays 20-0-18 2002 Oakland Athletics 20-0-10
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In a season, the 1943 Boston Red Sox--31 extra inning games, according to a Red Sox fan website.
Detroit Tigers, from September 9-13, 1908, played in 5 consecutive extra-inning games, winning 2 and losing 3.