How entangled can a multi-party system possibly be?

The geometric measure of entanglement of a pure quantum state is defined to be its distance to the space of pure product (separable) states. Given an n-partite system composed of subsystems of dimensions d1,…,dn, an upper bound for maximally allowable entanglement is derived in terms of geometric measure of entanglement. This upper bound is characterized exclusively by the dimensions d1,…,dn of composite subsystems. Numerous examples demonstrate that the upper bound appears to be reasonably tight.