If your opponent uses the ability where you can't untap until next turn can he use another card like that and have it stack for multiple turns?
It depends on the exact wording of the card in question (it's always helpful to give specific examples in these types of questions). If you mean a card like sleep, then no, it will not stack. Sleep makes the creatures not untap during the opponent's next untap step; even if you cast 2 of them on the same turn, they would both affect the same untap step.
But if you mean something like yosei, the morning star, then 2 instances of his triggered ability would stack. Each one would cause your opponent to skip his next untap step; so there are 2 replacement effects trying to replace the untap step. One if them would replace the next untap step with skipping that step; the other would would then still be around, waiting to skip the untap step that would happen after that.