What is the sum of the smallest and the greatest numbers formed using the digits 6, 5, 9 and 3 repeating any number of times?

To form the greatest number, we want the largest digits in the highest place values.
The largest digit available is 9. To make the number as large as possible, we repeat 9 twice, so it appears three times.
The digits to use are 9, 9, 9, 6, 5, 3.
Arranging these digits in descending order gives the greatest number:

$$999653$$

To form the smallest number, we want the smallest digits in the highest place values.
The smallest digit available is 3. To make the number as small as possible, we repeat 3 twice, so it appears three times.
The digits to use are 3, 3, 3, 5, 6, 9.
Arranging these digits in ascending order gives the smallest number:

$$333569$$

We add the two numbers found in Step 1 and Step 2.

$$999653 + 333569$$ $$999653 + 333569 = 1333222$$

The sum of the smallest and the greatest numbers is $\text{1333222}$.