We say that a space $X$ admits a homology exponent if there exists an exponent for the torsion subgroup of $H^*(X;\Z)$. Our main result states that if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form $B\Z/2^r$, $S^1$, $\CP^\infty$, and $K(\Z,3)$, or it has infinitely many non-trivial homotopy groups and k-invariants. Relying on recent advances in the theory of H-spaces, we then show that simply connected H-spaces whose mod $2$ cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod $2$ finite H-spaces with copies of $\CP^\infty$ and $K(\Z,3)$.