The New England Patriots had won 21 in a row if you include regular and post season -2003-2004 season.
Regular season only The New England Patriots have 18 Consecutive wins in the 2003-2004 season
Chicago Bears had 18 regular season straight wins 1933-34 & 1941-42 as well as;
Cleveland Browns had 18 -1947-48
Miami Dolphins had 18 -1972-73
San Francisco had 18 -1989-90
Denver Broncos had 18-1997-98
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The football league with the most consecutive title wins at the start of season is the Manchester United. These were in the years 1998 - 1999, 1999 - 2000, 2000 - 2001, 2006 - 2007, 2007 - 2008, and 2008 - 2009.