Where AR is the set of arithmetic Turing degrees, 0^{(\omega}) is the least member of {\mathbf{\alpha}^{(2)} |\mathbf{a} is an upper bound on \mathrm{AR}}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on \mathbf{HYP}, whose hyperjump is the degree of Kleene's \mathcal{O}. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In \S1 we review the basic definitions from [3] which are needed to state the general results.