What are "non-Keplerian" orbits? What are some familiar examples in our solar system, and can some still be closed? This excellent answer to Forms of stellar orbits around the galactic center invokes the following concepts:

non-Keplerian orbits
closed orbits

I have a fairly good idea what these mean and so might many of us, but our ideas may not overlap completely.
For the sake of future stack exchangers and orbiteers, how can we best define these?
Question(s):

What exactly are &quot;non-Keplerian&quot; orbits?
What are some familiar examples of distinctly non-Keplerian orbits in our solar system?
What are closed orbits?
What are some familiar examples of distinctly non-closed orbits in our solar system?
Can some non-Keplerian orbits still be closed?

Included should be the understanding that we're talking about soft limits here; no orbit is exactly Keplerian because gravity goes everywhere (there are no &quot;gravitational dielectrics&quot; or opposite charges for field lines to terminate). But for practical purposes we can treat many orbits as Keplerian for many purposes, and some we can't at all.
What exactly are &quot;non-Keplerian&quot; orbits?

Orbits that don't follow Kepler's laws.
Strictly speaking, all orbits are non-Keplerian. In practice, one can model some orbits as basically Keplerian, but with perturbations. Sun synchronous satellites are one example of orbits that are close to Keplerian, but not quite so. The Earth's equatorial bulge makes satellites fail to orbit in a plane. Sun synchronous satellites take advantage of the precession induced by the Earth's equatorial bulge so that their orbits precess by 360° in one year. Connor Garcia's example of satellites in pseudo-orbits about the Sun-Earth linear Lagrange points form another set of examples.

What are some familiar examples of distinctly non-closed orbits in our solar system?

Every planet, every moon, every asteroid, ...

What are closed orbits?

Bound orbits are orbits that stay bound to some central object. Parabolic and hyperbolic trajectories are not bound. Closed orbits are bound orbits that repeat their path. Any attractive central force law can result in circular orbits. These are trivially closed. There are only two attractive central force laws that can result in closed noncircular orbits: An inverse square law (e.g., Newtonian gravitation), and a linear force law (e.g., a Hookean spring). This is Bertrand's Theorem.

What are some familiar examples of distinctly non-Keplerian orbits in our solar system?

Every planet, every moon, every asteroid, ...

Can some non-Keplerian orbits still be closed?

Other than a Hookean spring, no.