[gumpzilla]

You're being pretty vague in your requirements, but the following series of 6 games can ensure that everybody plays three games and plays everybody at least once and one player exactly twice:

ABC
ADE
AFB
BDE
CDF
CEF

The problem you're running into is because you insist that two games be played simultaneously so that all players are used. This can't work: once you have ABC, ADE, and AFB (I arbitrarily pick B as the person A plays twice, it doesn't matter what you name him), this forces BDE to make sure that B plays with the two people he hasn't yet, CDF and CEF are necessary for D and E respectively.