I'm not following you here as far as upgrading dice. If you run out of dice to flip when you upgrade, you add a die.
I'm not aware of any RAW hard cap on dice use.
See pages 102 and 390.
Page 102 describes skill ranks up to 5. It describes ranks 1 and 2 as skilled, but a beginner. Ranks 3 and 4 are "professionals". Then, rank 5 is described as rarely obtained and such skilled people being known far and wide for such. It doesn't describe a skill rank 6 or beyond.
Page 390 describes Minions as having one skill rank below the current number in its group. So, a group of 6 provides the maximum group skill potential.
Thus, very large Minion groups (beyond 6) aren't going to benefit from offensive ability, but will still greatly benefit from defensive ability - more Soak and will not suffer offensive ability until there total number drops below 6.
Like I said, you could easily have Minion groups numbering 60 and not violate any rules. But, be aware that their offensive ability will not increase above that of a like group of 6, which could cause some realism issues. Ten groups of 6 Stormtroopers each are going to be much, much, much more powerful (10 attacks) then one group of 60 (1 attack of the same power as the smaller groups), even though they both represent 60 Stormtroopers.