This paper analyzes the performance of distributed Medium Access Control (MAC) protocols in ultra-dense multichannel wireless networks, where N frequency bands (or channels) are shared by M = mN devices, and devices make decisions to probe and then transmit over available frequency bands. While such a system can be formulated as an M-player Bayesian game, it is often infeasible to compute the Nash equilibria of a large-scale system due to the curse of dimensionality. In this paper, we exploit the Mean Field Game (MFG) approach and analyze the system in the large population regime (N tends to ∞ and m is a constant). We consider a distributed and low complexity MAC protocol where each device probes d/k channels by following an exponential clock which ticks with rate k when it has a message to transmit, and optimizes the probing strategy to balance throughput and probing cost. We present a comprehensive analysis from the MFG perspective, including the existence and uniqueness of the Mean Field Nash Equilibrium (MFNE), convergence to the MFNE, and the price of anarchy with respect to the global optimal solution. Our analysis shows that the price of anarchy is at most one half, but is close to zero when the traffic load or the probing cost is low. Our numerical results confirm our analysis and show that the MFNE is a good approximation of the M-player system. Besides showing the efficiency of the considered MAC for emerging applications in ultra-dense multichannel wireless networks, this paper demonstrates the novelty of MFG analysis, which can be used to study other distributed MAC protocols in ultra-dense wireless networks.