Note - I think every single one of these involves either Julio Franco or Terry Mulholland.
http://www.baseball-reference.com/oracle/linkable_most.shtml
Naturally, guys who play towards the middle of the century are the most-linked. I'd think it would help to play for a number of teams, also...and, looking at mid-century, even to play for 2-3 teams during the reserve system was probably a lot. Thirty years from now though...Franco is definitely at the top of the list.
How about this one: Lou Boudreau to Grady Sizemore
Better:
Tris Speaker - George Uhle (25 Indians)
George Uhle - Bob Feller (36 Indians)
Bob Feller - Minnie Minoso (49 Indians)
Minnie Minoso - Thad Bosley (80 White Sox)
Thad Bosley - Juan Gonzalez (89 Rangers)
Juan Gonzalez - Grady Sizemore (05 Indians)
Edited to note: The irony is that Gonzalez and Grady never actually played at the same time, as it was Juan's injury that paved the way for Grady to start on the team. Excluding Gonzo results in a Minoso-Brian Downing-Jose Hernandez chain.