We define an extremal computer as one which uses physical resources as effectively as possible for computation. As far as presently understood, extremal computers should be molecular computers. A number of computing mechanisms at the cellular level are considered, including word processing, kinetic mechanisms, conformational mechanisms, control processes in the membrane, diffusional mechanisms, and mechanisms involving sound waves. The hypothesis is put forward that the cyclic nucleotide system of intraneuronal information processing serves both as an analog diffusional mechanism and as a link between extremal (molecular) computing processes and macroscopic computing processes involving the nerve impulse. According to this hypothesis the computing power of the brain is primarily based on intracellular processes. The describability of nature by (computable) physical laws is limited by the extreme of computing which is possible in the physical universe. The paradox of molecular computing is that the unpredictable features exhibited by an object system which is not computable by an extremal computer could not be distinguished from new computing primitives which would allow more extremal computing. The authors present separate possible resolutions of this paradox. But under any interpretation the limits on the describability of nature imposed by the physical limits of computing imply that macromolecular processes of the type which occur in living systems provide a more fundamental foundation stone for the structure of science and of nature than previously thought.