  We study maximally multipartite entangled states in the context of Gaussian
continuous variable quantum systems. By considering multimode Gaussian states
with constrained energy, we show that perfect maximally multipartite entangled
states, which exhibit the maximum amount of bipartite entanglement for all
bipartitions, only exist for systems containing n=2 or 3 modes. We further
numerically investigate the structure of these states and their frustration for
n<=7.
