If $G$ is a compact group, and we choose the normalized Haar measure $\mu$ on $G$, then $L^2(G)$ is a Banach algebra with convolution. Indeed, if $f$ and $h$ are in $L^2(G)$, then we use Holder at the second step to compute
\begin{align*} |f\ast h(x)|\leq \int_G|f(y)h(x-y)|1\ d\mu(y)\leq \|fh\|_2\|1\|_2\leq \|f\|_2\|h\|_2.
\end{align*}
Thus 
$$\|f\ast h\|_2\leq \|f\ast h\|_{\infty}\leq \|f\|_2 \|h\|_2.$$
This Banach $\ast$-algebra is not only reflexive, but it is also a Hilbert space.