We report on a new class of elastic architected materials with hybrid unit cells, consisting of discrete elastic elements with non-convex strain energy and one convex (but possibly nonlinear) elastic element, to obtain a reversible multifunctional material with extreme energy dissipation. The proposed design exploits numerically optimized nonlinearities in the force–displacement response of the sub-unit-cell elements to approach the theoretical limit of specific damping capacity in any material, Psi_th=8. Specific damping capacities up to Psi=6.02 were experimentally demonstrated, which are far greater than any experimental value previously reported, including in high damping elastomers (Psi<4.5). Remarkably, this damping performance is achieved even with a single unit cell, thus avoiding the need for thick multi-cell designs. Furthermore, the proposed design offers relatively high stiffness and low transmitted stress upon compression. The proposed concept could enable the design of reversible impact-resistant structures with superior crashworthiness and energy dissipation.