What is the greatest number of acute angles that a right triangle can contain?

The greatest number is the least number and the exact number: two.

Every right triangle contains one right angle, that's what makes it a right triangle.

The other two angles are acute angles. We know that because they are complementary angles, summing to a right angle. Since each angle has a positive measure, each must be less than a right angle. So each must be acute.

Answer: 2