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Riggs Stephenson and Bill Madlock are tied for the Cubs career batting average leader with .336.
Chicago Cubs
The best career average for a major leaguer was .367 (Ty Cobb)
Rose got hit 4191 against the Chicago Cubs on September 8, 1985.
As far as batting average goes that would be Hall of Famer Rogers Hornsby who played with the Cardinals, Giants, Braves, Cubs, and Browns between 1915-1937. His career batting average was .358 and he is second all time in career batting average behind Ty Cobb.