A while back Daniel asked if there was a limit to the number of players and minor league affiliates an MLB team can have. There are limits at each level but other ways to store players.
Minor league rosters according to MiLB.
AAA -25 active players
AA – 25 active players
A+ – 25 active players (35 players under control on roster)
A – 25 active players (35 players under control on roster)
Rookie (Pioneer League) – 35 players on active roster
AZL – 35 players on active roster. Dodgers have two teams
DSL – 35 players on active roster.