How about five outs:
Bases loaded, 1 out. Batter hits to deep right and runners take off. Right fielded makes an amazing catch while R2 from second and R3 from third cross the plate assuming that the catch was missed while R1 stops at second and attempts to return to first to tag up. R2 and R3 crossed the plate before the appeal out on R1 which would have resulted in their run counting. Right fielder throws to first for the third out. Second-baseman calls for the ball to appeal R2 leaving early, umpire calls R2 out (4th out) and removes his run. F5 calls for the ball, F4 throws it to him and F5 appeals on R3 leaving early. Umpire calls R3 out (5th out), removes his run, and then prepares to eject the offensive coach who is about to have a total meltdown.
I think it would also be possible to get 6 outs: bases loaded, batter singles and gets caught in a run-down between first and second. During the run-down, R3, R2, and R1 all run home and all miss the plate. Then the batter is tagged for the third out. The catcher calls for the ball and stands on the plate. He looks at the umpire and says, "he missed the plate." The umpire says, "which one?" The catcher says, "the guy from first." Umpire, "yup, OUT." The catcher then says, "the guy from second also missed the plate." Umpire, "that's true too, he's OUT!" The catcher pauses and says, "Mr. Umpire, I also think the guy on third skipped over the plate." Umpire says, "sounds right to me, he's OUT!!! No runs score!"