  A regenerative composition structure is a sequence of ordered partitions
derived from the range of a subordinator by a natural sampling procedure. In
this paper, we extend previous studies Barbour and Gnedin (2006), Gnedin,
Iksanov and Marynych (2010) and Gnedin, Pitman and Yor (2006) on the
asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in
the case when the L{\'e}vy measure of the subordinator has a property of slow
variation at 0. Using tools from the renewal theory the limit laws for $K_n$
are obtained in terms of integrals involving the Brownian motion or stable
processes. In other words, the limit laws are either normal or other stable
distributions, depending on the behavior of the tail of L{\'e}vy measure at
$\infty$. Similar results are also derived for the number of singleton blocks.
