Why is the Moon's orbit so complicated? I like to think I have a good enough understanding of astrophysics but there's still one thing that I just can't explain. 

Why does the Moon orbit the Earth the way it does?

Many other moons in the Solar System orbit their planet along the equator and have a nearly circular orbit. After researching, it seems to boil down to two main causes:


Since every planet rotates on its axis, it has an equatorial bulge, and that extra mass around its equator creates an unequal gravitational field that over millions of years aligns the satellite's orbit around it.
There's a phenomenon called tidal circularization, which on long timescales, acts to dampen the satellite's eccentricity and make its orbit nearly circular. 


Okay, that makes sense for most moons. But our own Moon has an orbit that is notably eccentric and it's also inclined to Earth's equator by an amount that somehow changes over time from 18° to 28°. Why is the Moon's orbit so complicated?
The moon is so big that the processes that circularize and reduce the equatorial inclination would take much longer. The moon is big because of how it formed: a huge collision in the early solar system. (Unlike, say the Galilean moons that probably formed along with Jupiter, or Triton, that looks like a captured TNO)

The other fact that makes its orbit complex is that the moon is the closest moon to the sun, so solar perturbation is more significant than for other moons.  Solar perturbation is the main cause of the periodic variation in inclination and the various other forms of precession seen in the moon's orbit.

The sun also pulls the moon out of a equatorial orbit and into a roughly ecliptic orbit as the perturbations of the sun become more significant than those of the equatorial bulge. As noted elsewhere  at lunar distance, the sun provides more than 50% of the gravitational field.

So the reason the moon's orbit is complicated is that it does follow the laws of physics, and with three (or more) bodies, those laws have complex effects.