The trick is to pick as many teams as possible. If you can get each player to be on 7 teams (42 teams total), then each player will have played with all but 2 of the other players. But doing that certainly isn't easy (while holding to your criteria). Picking teams at random, I get to about 32 teams and no other teams are possible, even though most players have a lot of players that they haven't played with yet.

If I stick to having each player being on the same number of teams, then I usually get stuck after each player has been on two teams.

My quick look for some systematic way of choosing teams to maximize the number of teams possible also got stuck after two teams each. I think it is probably possible to get to 3 teams per player, while I suspect that it will be possible to prove that 4 teams per player is impossible.

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