DNA is renowned for its double helix structure and the base pairing that enables the recognition and highly selective binding of complementary DNA strands. These features, and the ability to create DNA strands with any desired sequence of bases, have led to the use of DNA rationally to design various nanostructures and even execute molecular computations. Of the wide range of self-assembled DNA nanostructures reported, most are one- or two-dimensional. Examples of three-dimensional DNA structures include cubes, truncated octahedra, octohedra and tetrahedra, which are all comprised of many different DNA strands with unique sequences. When aiming for large structures, the need to synthesize large numbers (hundreds) of unique DNA strands poses a challenging design problem. Here, we demonstrate a simple solution to this problem: the design of basic DNA building units in such a way that many copies of identical units assemble into larger three-dimensional structures. We test this hierarchical self-assembly concept with DNA molecules that form three-point-star motifs, or tiles. By controlling the flexibility and concentration of the tiles, the one-pot assembly yields tetrahedra, dodecahedra or buckyballs that are tens of nanometres in size and comprised of four, twenty or sixty individual tiles, respectively. We expect that our assembly strategy can be adapted to allow the fabrication of a range of relatively complex three-dimensional structures.