The original question asked about holding a kicker with 3 aces. Using AAA33 as the hand in question, here are the numbers:
holding AAA, p( Aces/kicker) = 0.93%, p(aces/no kicker) = 3.33%.
holding AAA3, P(Aces/kicker) = 2.13%, p(aces/no kicker) = 0.
There are differences in the full house and trips results, but they are dwarfed by the big hands. So, if you don't hold the kicker, you are trading 1.2% chance of hitting aces/kicker for a 3.33% chance of hitting aces.
By the way, this is the worst hand for holding a kicker since you are throwing away one kicker. The difference is still very large, about 21 coins.