Minnesota beat Michigan nine straight years from 1934 thorugh 1942. Only two other teams have a five-game win streak against Michigan, and those were Army from 1945-1954 (didn't play every year, obviously), and Purdue from 1962-1966. Five teams have four-game win streaks: Illinois (1950-1953); Michigan State (3 times, 1934-1937, 1950-1953, and 1959-1962); Minnesota (1960-1963); Notre Dame (1987-1990); and Ohio State (3 times, 1934-1937, 1960-1963, 2004-2007).
Note: Cornell actually has seven-game win streak, but that was 1889-1894, with multiple game played in 1891 and 1892. I think the game had changed significantly from the last of those Cornell wins to the earliest of any of the wins previously mentioned, so I don't include Cornell's streak.