Spatially extended dynamical systems exhibit complex behaviour in both space and time--spatiotemporal chaos. Analysis of dynamical quantities (such as fractal dimensions and Lyapunov exponents) has provided insights into low-dimensional systems; but it has proven more difficult to understand spatiotemporal chaos in high-dimensional systems, despite abundant data describing its statistical properties. Initial attempts have been made to extend the dynamical approach to higher-dimensional systems, demonstrating numerically that the spatiotemporal chaos in several simple models is extensive (the number of dynamical degrees of freedom scales with the system volume). Here we report a computational investigation of a phenomenon found in nature, 'spiral defect' chaos in Rayleigh-Benard convection, in which we find that the spatiotemporal chaos in this state is extensive and characterized by about a hundred dynamical degrees of freedom. By studying the detailed space-time evolution of the dynamical degrees of freedom, we find that the mechanism for the generation of chaotic disorder is spatially and temporally localized to events associated with the creation and annihilation of defects.