  Let $\Gamma$ be a discrete group. We show that if $\Gamma$ is nonamenable,
then the algebraic tensor products $C^*_r(\Gamma)\otimes C^*_r(\Gamma)$ and
$C^*(\Gamma)\otimes C^*_r(\Gamma)$ do not admit unique $C^*$-norms. Moreover,
when $\Gamma_1$ and $\Gamma_2$ are discrete groups containing copies of
noncommutative free groups, then $C^*_r(\Gamma_1)\otimes C^*_r(\Gamma_2)$ and
$C^*(\Gamma_1)\otimes C_r^*(\Gamma_2)$ admit $2^{\aleph_0}$ $C^*$-norms.
Analogues of these results continue to hold when these familiar group
$C^*$-algebras are replaced by appropriate intermediate group $C^*$-algebras.
