The performance of amplify-and-forward (AF) multi-way relay networks (MWRNs) with very large relay antenna arrays is investigated. Asymptotic signal-to-interference-plus-noise ratio (SINR) expressions are derived when the number of antennas at the relay is substantially higher than the number of user nodes. These asymptotic SINRs are independent of the fast fading component of the wireless channel, and hence, the latency in the air interface due to fast fading can be reduced. The transmit power at each user node and/or relay can be scaled down inversely proportional to the number of antennas at the relay without degrading the system performance. The asymptotic SINR expressions are then used to derive asymptotic spectral and energy efficiencies of AF MWRNs with symmetric and asymmetric data traffic. These spectral and energy efficiency results show that MWRNs with massive MIMO achieve substantial multIPlexing gains while allowing the transmit powers of user nodes to become infinitesimal.