We introduce in this paper the recursive Hessian sketch, a new adaptive filtering algorithm based on sketching the same exponentially weighted least squares problem solved by the recursive least squares algorithm. The algorithm maintains a number of sketches of the inverse autocorrelation matrix and recursively updates them at random intervals. These are in turn used to update the unknown filter estimate. The complexity of the proposed algorithm compares favorably to that of recursive least squares. The convergence properties of this algorithm are studied through extensive numerical experiments. With an appropriate choice or parameters, its convergence speed falls between that of least mean squares and recursive least squares adaptive filters, with less computations than the latter.