I think the MLB record is around 20. I figure a great team could have a 65% chance of winning a random game, which makes even 3 in a row improbable. Of course, sweeps happen usually because a very good team is playing a rather poor team. But once you go to double digits, ....
The funny thing is the Braves are still 4 GB.
This is my problem with extended playoffs, the best team is unlikely to win out.