Ok so here are my thoughts. I wanted to construct something like a clock where each circular sector is made of a different material. The incoming angle of the light ray with respect to the boundary of the first to the second layer is $\alpha$ (see image). To make things easy I wanted to choose the second material and circular section in such a way, that the incoming angle in the second refraction is still $\alpha$. If the refractive index of the second material is bigger than the index of the first material, this means that the circular sector of the second material must be smaller $\alpha - \epsilon$. Since the sum of the angles in a triangle is $\pi$ we find that the outgoing angle $\beta$ must be \begin{equation} \beta + \pi/2 + \pi/2 - \alpha + \alpha - \epsilon = \pi \quad \Rightarrow \quad \beta = \epsilon. \end{equation} Hence we must choose two materials with \begin{equation} \frac{\sin \alpha}{\sin \beta} = \frac{\sin \alpha}{\sin \epsilon} = \frac{n_2}{n_1} \end{equation} For the next circular sector, we proceed in the exact same way. The sector has the angle $\alpha - 2 \epsilon$ and the incoming angle is still $\alpha$, thus the ratio $\frac{n_3}{n_2}$ is equal to $\frac{n_2}{n_1}$. 586
This process can be repeated until we got around the entire circle. Important is that we have to choose $\alpha$ and $\epsilon$ in a way that we get around the entire circle, i.e. \begin{equation} \sum_n (\alpha - n \epsilon) \ge 2\pi. \end{equation} For instance choose $\alpha = 30^\circ$ and $\epsilon = 1^\circ$.
This does not entirely answer your question though, as the loop is not closed. The distance to the center of the circle gets larger and larger with every refraction. In the ideal case $\epsilon = 0$ you would have a closed-loop but for that, you would need a material with an infinite refractive index. So for a setup like this, we have shown that it is not possible to construct a closed loop. And even if I did not show it I predict that it is not possible at all (but I am happy to be convinced of the opposite).
Also notice, that even if you are happy with a loop that is not closed, you can not build such a device in real life as you run out of materials with a high enough refractive index. For $\alpha = 30^\circ$ and $\epsilon = 1^\circ$ you would already require $n_2/n_1 = 29$ which is impossible to achieve in real life (as far as I know). 586