We consider communication through an infinite cascade of identical discrete memoryless channels. We allow the source and destination nodes to use coding schemes of arbitrary complexity, but restrict the intermediate (relay) nodes to process blocks of a fixed blocklength. We calculate the optimal end-to-end rate, maximized over all possible processings at the relays, and show that it coincides with the end-to-end zero-error capacity. The optimal processing is shown to be identical at each relay and to correspond to a zero-error code. We also show that the rate of convergence to the asymptotic value is exponential in the length of the cascade.