Well, since there was no restriction on how many..
games were played, I guess it's possible to have a final inning win where 4 consecutive singles (that only advance each player by one base) and on the forth single the game winning run crosses the plate without any outs having been recorded. That same team, at their next game, in the 1st inning gets 3 consecutive singles (loading the bases). Now that is 7 consecutive singles, 1 run scored and no outs.
The next batter is where it gets tough to recognize what comes first, the hit or the run. If the batter gets a hit, is it recorded as a hit before the 3rd base runner crosses the plate, or vice versa? Are 8 consecutive hits possible?
But I'm now going with the 7.
Jim