  We introduce the forward limit set $\Lambda$ of a semigroup $S$ generated by
a family of substitutions of a finite alphabet, which typically coincides with
the set of all possible s-adic limits of that family. We provide several
alternative characterisations of the forward limit set. For instance, we prove
that $\Lambda$ is the unique maximal closed and strongly $S$-invariant subset
of the space of all infinite words, and we prove that it is the closure of the
set of images of all fixed points under $S$. It is usually difficult to compute
a forward limit set explicitly; however, we show that, provided certain
assumptions hold, $\Lambda$ is uncountable, and we supply upper bounds on its
size in terms of logarithmic Hausdorff dimension.
