R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both $\alpha$-recursive functionals and weak $\alpha$-recursive functionals for all admissible ordinals $\alpha$ such that $\lambda < \alpha^\ast$, where $\alpha^\ast$ is the $\Sigma_1$-projectum of $\alpha$ and $\lambda$ is the $\Sigma_2$-cofinality of $\alpha$. The theorem is also established for the metarecursive case, $\alpha = \omega_1$, where $\alpha^\ast = \lambda = \omega$.