Oakland Athletics with 5 between 1971-1975. This was before the playoffs were expanded to include an ALDS, which started in 1995. Since 1995 and the beginning of the ALDS, the New York Yankees went to 4 straight between 1998-2001.
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New York Yankees with 5 from 1949-1953 (won all 5) and 1960-1964 (won 2 and lost 3).