Assuming an astronomical object is a member of a two-body system and
has a spin axis and an orbital plane, can the object's spin axis
precess around an axis /other/ than the normal to its orbital plane?

Doesn't seem likely, not with only two bodies.

Okay, add a few more bodies.

Take the Moon, for example.  Its spin axis precesses about the normal
to its orbital plane.  

Why couldn't its spin axis also precess around the normal to the
ecliptic?

Or does it, but to a very much smaller degree?

Is that what nutation is?  A smaller and faster precession around some
other axis superposed on a somewhat larger and slower "main"
precession?
--
Dave