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2 at most.
Split the balls into 3 groups, 3:3:2
Weigh the 2 groups of 3 against each other, if it is even then you know the heaviest ball is in the group of 2. Therefore you can just weigh the 2 against each other to give you the heaviest one.
If when you weigh the two groups of 3 against each other and one is heavier than the other, you know which group he heavier ball is in. Now split the heavier group of 3 into individual balls and weigh 2 against each other. Is they are even, the 3rd ball is the heavier, if it is uneven, you will be able to see which is heavier