  We initiate the study of graph classes of power-bounded clique-width, that
is, graph classes for which there exist integers $k$ and $\ell$ such that the
$k$-th powers of the graphs are of clique-width at most $\ell$. We give
sufficient and necessary conditions for this property. As our main results, we
characterize graph classes of power-bounded clique-width within classes defined
by either one forbidden induced subgraph, or by two connected forbidden induced
subgraphs. We also show that for every positive integer $k$, there exists a
graph class such that the $k$-th powers of graphs in the class form a class of
bounded clique-width, while this is not the case for any smaller power.
