The largest integer that is not the sum of one or more distinct squares is 128 and is called the threshold of completeness for the set of squares. A natural extension is to consider positive cubes and higher powers. Previous research solved this problem for powers up to 7. Our contribution is the threshold of completeness for 8th powers as well as lower bounds for powers from 9 to 16. We also describe the mathematical methods we used to speed up computations. Using 200,000 computer cores, our methods could find the threshold of completeness for 9th powers in about a month, and for 10th powers in about a year.