DNA base-pairing is the central interaction in DNA assembly. However, this simple four-letter (A-T and G-C) language makes it difficult to create complex structures without using a large number of DNA strands of different sequences. Inspired by protein folding, we introduce hydrophobic interactions to expand the assembly language of DNA nano technology. To achieve this, DNA cages of different geometries are combined with sequence-defined polymers containing long alkyl and oligoethylene glycol repeat units. Anisotropic decoration of hydrophobic polymers on one face of the cage leads to hydrophobically driven formation of quantized aggregates of DNA cages, where polymer length determines the cage aggregation number. Hydrophobic chains decorated on both faces of the cage can undergo an intrascaffold "handshake" to generate DNA-micelle cages, which have increased structural stability and assembly cooperativity, and can encapsulate small molecules. The polymer sequence order can control the interaction between hydrophobic blocks, leading to unprecedented "doughnut-shaped" DNA cage-ring structures. We thus demonstrate that new structural and functional modes in DNA nanostructures can emerge from the synergy of two interactions, providing an attractive approach to develop protein-inspired assembly modules in DNA nanotechnology.