In this paper we study the limiting distribution of the maximum term of non-negative integer-valued moving average sequences of the form X n = Σ∞ i= - ∞β i o Z n - i where {Z n } is an iid sequence of non-negative integer-valued random variables with exponential type tails of the form 1 - F(n) ∼ Kn ζ (1 + λ) -n when n → ∞, ζ ∈ R, K, λ > 0, and o denotes binomial thinning. Several models are considered allowing different dependence structures of the thinning operations. For these models results are established which present similarities with those obtained for the classic linear Fmoving average: {X n } behaves as if it was iid regarding the limiting distribution of the maximum term. The paper concludes with some examples which apply the results to a particular model, the INAR(1), and with a simulation study to illustrate the results.