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Since the invention of polymer networks such as cross-linked natural rubber in the 19th century, it has been a dogma that stiffer networks are less stretchable. We report a universal strategy for decoupling the stiffness and extensibility of single-network elastomers. Instead of using linear polymers as network strands, we use foldable bottlebrush polymers, which feature a collapsed backbone grafted with many linear side chains. Upon elongation, the collapsed backbone unfolds to release stored length, enabling remarkable extensibility. By contrast, the network elastic modulus is inversely proportional to network strand mass and is determined by the side chains. We validate this concept by creating single-network elastomers with nearly constant Young’s modulus (30 kilopascals) while increasing tensile breaking strain by 40-fold, from 20 to 800%. We show that this strategy applies to networks of different polymer species and topologies. Our discovery opens an avenue for developing polymeric materials with extraordinary mechanical properties.