The Tower of Hanoi problem is generalized in such a way that the pegs are located at the vertices of a directed graph G, and moves of disks may be made only along edges of G. Leiss obtained a complete characterization of graphs in which arbitrarily many disks can be moved from the source vertex S to the destination vertex D. Here we consider graphs which do not satisfy this characterization; hence, there is a bound on the number of disks which can be handled. Denote by g_n the maximal such number as G varies over all such graphs with n vertices and S, D vary over the vertices. Answering a question of Leiss [Finite Hanoi problems: How many discs can be handled? Congr. Numer. 44 (1984) 221–229], we prove that g_n grows sub-exponentially fast. Moreover, there exists a constant C such that [Formula: see text] for each n. On the other hand, for each ɛ>0 there exists a constant C_ɛ>0 such that [Formula: see text] for each n.