Nanoparticles can be combined with nucleic acids to programme the formation of three-dimensional colloidal crystals where the particles' size, shape, composition and position can be independently controlled. However, the diversity of the types of material that can be used is limited by the lack of a general method for preparing the basic DNA-functionalized building blocks needed to bond nanoparticles of different chemical compositions into lattices in a controllable manner. Here we show that by coating nanoparticles protected with aliphatic ligands with an azide-bearing amphiphilic polymer, followed by the coupling of DNA to the polymer using strain-promoted azide-alkyne cycloaddition (also known as copper-free azide-alkyne click chemistry), nanoparticles bearing a high-density shell of nucleic acids can be created regardless of nanoparticle composition. This method provides a route to a virtually endless class of programmable atom equivalents for DNA-based colloidal crystallization.