At least one example of dimension $2$ (but I don't have an example in other finite dimension). Take $X={\mathbb R}^2$ with $\ell^1$-norm
$$\|x\|_1=|x_1|+|x_2|.$$
Then $X^*={\mathbb R}^2$ has the $\ell^\infty$-norm
$$\|y\|_\infty=\max(|y_1|,|y_2|).$$
I turns out that
$$\|x\|_1=\max(|x_1+x_2|,|x_1-x_2|)$$
and thus $X^*$ is isometric to $X$, *via* $x\mapsto(x_1+x_2,x_1-x_2)$.