In fact, there is not, unless we count objects such as this



which look like triangles touching at their corners. The reason there are none is that 2 successive angles of 60 degrees to the right (or left) form a triangle when the side lengths are equal, so our only constructions will either be a zig-zag like this



which doesn't close in on itself, or a triangle, which is regular.

For which n does there exist a _regular_ polygon with n sides?