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.