Although this question has been answered many times, I would like to add the following simple way to generate examples. It is clear that in the plane any central force, no matter how complicated, generates circular orbits as special cases. There are many ways to distribute matter in three space to generate such a force field. For example, any distribution of mass (which can, of course, consist of point masses) along the $z$-axis will produce such a field in the $x,y$-plane. As a side remark, such configurations occur in practice---e.g., as the gravitational field generated by a wire.
Although this question has been answered many times, I would like to add the following simple way to generate examples. It is clear that in the plane any central force, no matter how complicated, generates circular orbits as special cases. There are many ways to distribute matter in three space to generate such a force field. For example, any distribution of mass (which can, of course, consist of point masses) along the $z$-axis will produce such a field in the $x,y$-plane. As a side remark, such configurations occur in practice---e.g., as the gravitational field generated by a wire.