This paper studies the formation stabilization problem of asynchronous nonlinear multi-agent systems (MAS) subject to parametric uncertainties, external disturbances and bounded time-varying communication delays. A self-triggered min-max distributed model predictive control (DMPC) approach is proposed to handle these practical issues. At triggering instants, each agent solves a local min-max optimization problem based on local system states and predicted system states of neighbors, determines its next triggering instant and broadcasts its predicted state trajectory to its neighbors. As a result, the communication load is greatly alleviated while retaining robustness and comparable control performance compared to periodic algorithms. In order to handle time-varying delays, a novel consistency constraint is incorporated into each local optimization problem to restrict the deviation between the newest predicted states and previously broadcast predicted states. Consequently, each agent can utilize previously predicted states of its neighbors to achieve cooperation in the presence of time-varying delays and asynchronous communication induced by the distributed triggered scheduler. The recursive feasibility of the proposed algorithm and the closed-loop stability of MAS at triggering time instants are proven. Finally, numerical simulations are conducted to verify the efficiency of the proposed control method.