The anti-Waring problem considers the smallest positive integer such that it and every subsequent integer can be expressed as the sum of the k^th powers of r or more distinct natural numbers. We give a generalization that allows elements from any nondecreasing sequence, rather than only the natural numbers. This generalization is an extension of the anti-Waring problem, as well as the idea of complete sequences. We present new anti-Waring and generalized anti-Waring numbers, as well as a result to verify computationally when a generalized anti-Waring number has been found.