In this paper, we demonstrate a molecular system for the firstactiveself-assemblylinearDNA polymer that exhibits programmable molecular exponential growth in real time, also the first to implement “internal” parallel insertion that does not rely on adding successive layers to “external” edges for growth. Approaches like this can produce enhanced exponential growth behavior that is less limited by volume and external surface interference, for an early step toward efficiently building two and three dimensional shapes in logarithmic time. We experimentally demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism and results in the exponential growth of a population of polymers per unit time. In thesupplementary material, we note that an “extension” beyond conventional Turing machine theory is needed to theoretically analyze exponential growth itself in programmable physical systems. Sequential physical Turing Machines that run a roughly constant number of Turing steps per unit time cannot achieve an exponential growth of structure per time. In contrast, the “active” self-assembly model in this paper, computationally equivalent to a Push-Down Automaton, is exponentially fast when implemented in molecules, but istaxonomically less powerfulthan a Turing machine. In this sense, a physical Push-Down Automaton can bemore powerfulthan a sequential physical Turing Machine, even though the Turing Machine can compute any computable function. A need for an “extended” computational/physical theory arises, described in the supplementary material sectionS1.