A.S. Galbraith has communicated to us the following intriguing problem: does the completeness of a manifold imply, or is it implied by, the emptiness of the class $H^{2}B$ of bounded nonharmonic biharmonic functions? Among all manifolds considered thus far in biharmonic classification theory (cf. Bibliography), those that are complete fail to carry $H^{2}B$-functions, and one might suspect that this is always the case. We shall show, however, that there do exist complete manifolds of any dimension that carry...