We show that extreme value statistics are useful for studying the largest structures in the Universe by using them to assess the significance of two of the most dramatic structures in the local Universe – the Shapley supercluster and the Sloan Great Wall. If we assume that the Shapley concentration (volume ≈1.2 × 105 h−3 Mpc3) evolved from an overdense region in the initial Gaussian fluctuation field, with currently popular choices for the background cosmological model and the shape and amplitude σ8 of the initial power spectrum, we estimate that the total mass of the system is within 20 per cent of 1.8 × 1016 h−1 M⊙. Extreme value statistics show that the existence of this massive concentration is not unexpected if the initial fluctuation field was Gaussian, provided there are no other similar objects within a sphere of radius 200 h−1 Mpc centred on our Galaxy. However, a similar analysis of the Sloan Great Wall, a more distant (z∼ 0.08) and extended concentration of structures (volume ≈7.2 × 105 h−3 Mpc3), suggests that it is more unusual. We estimate its total mass to be within 20 per cent of 1.2 × 1017 h−1 M⊙ and we find that even if it is the densest such object of its volume within z= 0.2, its existence is difficult to reconcile with the assumption of Gaussian initial conditions if σ8 was less than 0.9. This tension can be alleviated if this structure is the densest within the Hubble volume. Finally, we show how extreme value statistics can be used to address the question of how likely it is that an object like the Shapley supercluster exists in the same volume which contains the Sloan Great Wall, finding, again, that Shapley is not particularly unusual. Since it is straightforward to incorporate other models of the initial fluctuation field into our formalism, we expect our approach will allow observations of the largest structures – clusters, superclusters and voids – to provide relevant constraints on the nature of the primordial fluctuation field.