Simultaneous wireless information and power transfer for multi-way relay networks with massive multiple-input multiple-output (MIMO) is investigated. By using two practically viable relay receiver designs, namely (i) the power splitting receiver and (ii) the time switching receiver, asymptotic signal-to-interference-plus-noise ratio (SINR) expressions are derived for an unlimited number of antennas at the relay. These asymptotic SINRs are then used to derive asymptotic symmetric sum rate expressions in closed-form. Notably, these asymptotic SINRs and sum rates become independent of radio frequency-to-direct current (RF-to-DC) conversion efficiency in the limit of infinitely many relay antennas. Moreover, the fundamental trade-off between the harvested energy and the achievable sum rate is quantified for both relay receiver structures. This analysis reveals that the transmit power of each user node can be scaled down inversely proportional to the number of relay antennas, and the corresponding asymptotic SINR and sum rate expressions become independent of the fast fading effects of the wireless channels. The presence of co-channel interference (CCI) can be exploited to be beneficial for energy harvesting at the relay, and consequently, the asymptotic harvested energy is an increasing function of the number of co-channel interferers. Nevertheless, the detrimental impact of CCI for signal decoding can be cancelled completely whenever the number of relay antennas grows without bound.