1906 and '07 Chicago Cubs

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1906: 116-36, .763; lost World Series
1907: 107-45, .704; won World Series

Starring the famous double play combination of shortstop Joe Tinker, second baseman Johnny Evers and first baseman Frank Chance as well as fellow Hall of Famer Mordecai "Three-Finger" Brown, the staff ace, the 1906 Cubs set a single-season record for winning percentage of .753 that still stands. The last time they lost consecutive games came on July 23 and 24. From that point on they went a bonkers 55-8, an .873 winning percentage. That stretch included winning streaks of 11 and 14 games and a period in which they won 37 of 39 games. They picked a bad time for their next two-game losing streak: Games 5 and 6 of the World Series, when the White Sox, their crosstown rivals, suddenly forgot they were known as the Hitless Wonders and put up eight runs in both games to win the title. 

With a virtually identical cast of characters the next year, the Cubs kept pace with their predecessors through 118 decisions; both went 87-31. From there, Chicago went "only" 18-10 (.643), but this time around, it won the championship, sweeping four straight games from the Tigers after a Game 1 tie. (The picture above shows Chance batting during that year's Fall Classic.)