The present paper investigates the achievable data rates of multi-hop amplify-and-forward multi-antenna relay channels with arbitrary number of hops K. Each multi-antenna terminal in the system amplifies the received signal and retransmits it upstream. To analyze the ergodic end-to-end mutual information of the system, one has to perform averaging over the fading coefficients. To overcome this difficulty we apply large-system analysis, based on the assumption that the number of antennas grows without bound at every terminal. Using the replica method, we derive an explicit asymptotic expression for the ergodic mutual information between the input and output of the K-hop channel with no restrictions on the channel inputs. Numerical results support the validity of the replica analysis and show that the result is tight even for small antenna arrays.