The New York Giants winning 26 in a row.

The 2002 A's would probably be next when they won 20 or so games back in '02.

I believe it was 21 by Oakland a few years back.

EDIT:

I stand corrected;
here is the correct answer:

http://www.baseball-almanac.com/recbooks...

The New York Giants in 1916 they are no longer a team though but they won 26 games the problem with this is though they tied a game in the streak so it is questionable. If you decided not to count that then your winner is the cubbies with 21 wins in 1935