To find the highest possible value of the average of the 20 integers, we need to maximize the sum of these integers. Let's consider the extremes of the numbers that do not exceed 5.

1. Identifying the extremes:
Since there are exactly 20 integers that do not exceed 5, the remaining 10 integers must be greater than 5. To maximize the sum, we want these 10 integers to be as large as possible. Therefore, the extremes of the integers that do not exceed 5 would be 0 and 5.

2. Finding the sum:
We know that the average of the 30 integers is 5, which means the sum of all 30 integers is 30 * 5 = 150. Since there are 10 integers greater than 5, their sum must be 150 - 20 * 5 = 50.

3. Maximizing the sum of the 20 integers:
To maximize the sum of the 20 integers, we want to distribute the remaining sum of 50 as evenly as possible among these 20 integers. Since the extremes are 0 and 5, we can distribute the remaining sum in such a way that each of the 20 integers is as close to 5 as possible.

If we divide the remaining sum of 50 equally among the 20 integers, each integer would have a value of 50/20 = 2.5. However, this is not possible, as the integers must be whole numbers. Therefore, we need to adjust the distribution.

To make the integers as close to 5 as possible, we can have 15 integers with a value of 5 and 5 integers with a value of 0. This way, the sum of the 20 integers would be 15 * 5 + 5 * 0 = 75.

4. Calculating the average:
The average of the 20 integers is given by the sum of these integers divided by 20. Therefore, the average would be 75/20 = 3.75.

However, we need to remember that the answer choices are given as whole numbers. Since the average cannot be a decimal, we need to round up to the nearest whole number. Therefore, the highest possible value of the average of these 20 integers is 4.5, which corresponds to option (D).