Question: Roughly how many kinds of closed or periodic orbits are there in the circular restricted three-body problem? Has anyone made an attempt to enumerate and classify/categorize them?
Notes: If there is a distinction between closed and periodic required to answer the question, you can choose either one. It is possible that an answer can be found in references contained in this answer.
The Circular, Restricted Three Body Problem or CR3BP or CRTBP is the mathematical problem where the orbit of a third object of negligible mass responds to the gravity field of two massive objects in circular orbit around a their center of mass.
Commonly people think of the two bodies as the Sun and Earth, or the Earth and the Moon in the approximation that there are no other bodies and these two are in circular orbits, and the third body is a spacecraft or small asteroid of negligible mass. Early work by the famous mathemagician Joseph-Louis Lagrange discussed the five equilibrium points or Lagrange Points where the acceleration on the third body is zero in the rotating frame.
The CR3BP is a purely mathematical problem and is helpful as a starting point for thinking about orbits of small bodies in realistic scenarios where the effect of two bodies dominate, such as a sun-planet or planet-moon system.
Most orbits of the third body in the CR3BP are chaotic and non-repeating (non-periodic or non-closed). However there are certain cases where the orbit are. After a certain period of time, the body will pass through its exact starting point with the same exact velocity vector and go on to repeat the same path. This is where the depth and complexity of the problem really begins.
See this answer to What sort of orbital elements are used to describe halo orbits? and have a gander at Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem and then answers to What sort of orbital elements are used to describe halo orbits?