I have Strionic Resonator on the battlefield and I cast Kozilek, the Great Distortion. I use the resonator to copy Kozilek's draw ability after it enters the stack. Do I draw twice as many cards, or does Kozilek's ability check to see my hand size after the copy resolves and sees I already have seven cards (which means I don't get to draw any additional cards from the ability resolving a second time)?
You will draw to 7 from the first Kozilek ability.
Then, you will have priority and the opportunity to use some of the cards drawn before the second ability resolves, which will refil you to seven.
For instance, if the first Kozilek ability draws you Lightning Bolt and you cast it, you will be at 6 cards when the second ability starts to resolve and so you'll draw back to 7.
The triggered ability has an 'if clause', and those clauses are checked both when the ability goes to the stack and when it tries to resolve. This is still true for the copy. If you don't do anything else, the copy will resolve first filling your hand to seven, and then when the original ability tries to resolve it sees you have seven cards and is removed from the stack.
However, remember that each object on the stack resolves separately, including copies. So if you have some instant-speed way of reducing your hand size (like casting an Instant from your hand), you can do that between the resolution of the copy and of the original ability.
You'll have two 'draw till you hold 7' triggers on the stack.
So when the first resolves you'll draw to 7. If you have any instants or ways of getting rid of cards, you can do so before the second trigger resolves - when it does you'll draw till 7 once more.
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Then, you will have priority and the opportunity to use some of the cards drawn before the second ability resolves, which will refil you to seven.
For instance, if the first Kozilek ability draws you Lightning Bolt and you cast it, you will be at 6 cards when the second ability starts to resolve and so you'll draw back to 7.
However, remember that each object on the stack resolves separately, including copies. So if you have some instant-speed way of reducing your hand size (like casting an Instant from your hand), you can do that between the resolution of the copy and of the original ability.
So when the first resolves you'll draw to 7. If you have any instants or ways of getting rid of cards, you can do so before the second trigger resolves - when it does you'll draw till 7 once more.