  We show that, if $w_1, \ldots , w_6$ are words which are not an identity of
any (non-abelian) finite simple group, then $w_1(G)w_2(G) \cdots w_6(G) = G$
for all (non-abelian) finite simple groups $G$. In particular, for every word
$w$, either $w(G)^6 = G$ for all finite simple groups, or $w(G)=1$ for some
finite simple groups. These theorems follow from more general results we obtain
on characteristic collections of finite groups and their covering numbers,
which are of independent interest and have additional applications.
