Are there any examples of uncountably infinite sets with cardinality strictly greater than c other than the power set of the set of real numbers and the sequence of strictly larger sets obtained by constructing a sequence of uncountably infinite sets by creating the power set of the predecessor set, beginning with the real numbers? I.e., with cardinality other than c,2c,22c,...?
This question is not a duplicate of the question "What infinity is greater than the continuum?" This question asks if there are any infinite cardinal numbers that are not elements of the countably infinite set {c,2c,22c,...}. Equivalently, does N is an infinite cardinal number imply N is an element of {c,2c,22c,...}?