Number of consecutive years managing the same MLB team: 50, by Connie Mack, Philadelphia Athletics, 1901 to 1950.
In case you meant by players only:
Number of consecutive no-hit, shutout, complete game victories pitched in a row: 2, by Johnny Vander Meer, 1938, for the Cincinnati Reds.
What is the greatest accomplishment you have ever seen in person at a sports stadium?