  Extremal clusters of stationary processes with long memory can be quite
intricate. For certain stationary infinitely divisible processes with
subexponential tails, including both power-like tails and certain lighter
tails, e.g. lognormal-like tails, such clusters may take the shape of stable
regenerative sets. In this paper we show that for semi-exponential tails, which
are even lighter, a new shape of extremal clusters arises. In this case each
stable regenerative set supports a random panoply of varying extremes.
