In this talk we discuss some extremal properties of the so-called max-INAR process of order one based on the binomial thinning operator, and marginal distribution exhibiting regularly varying right tail. In particular, attention is paid to the limiting distribution of the number of exceedances of high levels and the joint limiting law of the maximum and the minimum. Furthermore, we also look at the extremal behavior of the max-INAR process of order one under the assumption that its corresponding thinning parameter is random. Finally, the periodic case will be also addressed.