  Let $G$ be a non-compact group, $K$ the compact subgroup fixed by a Cartan
involution and assume $G/K$ is an exceptional, symmetric space, one of Cartan
type $E,F $ or $G$. We find the minimal integer, $L(G),$ such that any
convolution product of $L(G)$ continuous, $K$-bi-invariant measures on $G$ is
absolutely continuous with respect to Haar measure. Further, any product of
$L(G)$ double cosets has non-empty interior. The number $L(G)$ is either $2$ or
$3$% , depending on the Cartan type, and in most cases is strictly less than
the rank of $G$.
