  We say a graph $H$ is $r$-rainbow-uncommon if the maximum number of rainbow
copies of $H$ under an $r$-coloring of $E(K_n)$ is asymptotically (as $n \to
\infty$) greater than what is expected from uniformly random $r$-colorings. Via
explicit constructions, we show that for $H\in\{K_3,K_4, K_5\}$, $H$ is
$r$-rainbow-uncommon for all $r\geq {|V(H)|\choose 2}$. We also construct
colorings to show that for $t \geq 6$, $K_t$ is $r$-rainbow-uncommon for
sufficiently large $r$.
