  The article describes the limiting distribution of the extremes of
observations that arrive in clusters. We start by studying the tail behaviour
of an individual cluster and then we apply the developed theory to determine
the limiting distribution of $\max\{X_j: j=0,\ldots, K(t)\}$, where $K(t)$ is
the number of i.i.d. observations $(X_j)$ arriving up to the time $t$ according
to a general marked renewal cluster process. The results are illustrated in the
context of some commonly used Poisson cluster models such as the marked Hawkes
process.
