When does angular position, or angular displacement, not obey the rules of vector addition? The only examples I've found talk about rotating an object about one axis and then about another axis. When you reverse the order, the object ends up in a different position. However, as those rotations happen on different planes and at different times, I see them as unrelated, each with their own equation. Can anyone think of an example of when angular position or displacement not behave like vectors, when the rotation happens on a single plane?
 A: From the title of the question:

When does angular position, or angular displacement, not obey the rules of vector addition?

The answer to this question is simple: It's whenever the rotations span more than a two dimensional space. Vector addition must be commutative by definition: $a + b = b + a$ for all elements $a$ and $b$ in the space. As you've already noted in the question, rotations don't commute in three dimensional space. This is also true in four dimensional space, five dimensional space, and so on.

From the body of the question:

Can anyone think of an example of when angular position or displacement not behave like vectors, when the rotation happens on a single plane?

The answer to this question is also simple: "No". Rotations in two dimensional space do commute. In two dimensional space, performing rotation A followed by rotation B is exactly the same as performing rotation B followed rotation A.
A: In the comments, you clarified what you had in mind for a "vector" of rotation.

Well, in my mind the magnitude of angular position would be the counterclockwise angle from the positive x axis in the plane of rotation, to a ray from the origin to the rotating object. The direction would be the same as the direction of angular velocity in 3D space. So yes, what I had in mind would be a triple of angles: the first in the plane of rotation, and the next two in 3D.

This seems to be two different ideas:  Up until the last sentence, it looked like you were going to say that if you rotate counterclockwise in the $xy$ plane by angle $\theta$ that the "rotation vector" would be $(0,0,\theta)$  I cannot quite make sense of your last sentence, so I'm sticking with this interpretation.
If you restrict yourself to rotations in this plane and you also restrict yourself to rotations around a single point, I think this is going to turn out to work like vector addition, but in a very trivial way.  You've restricted yourself to a situation where all of your "vectors" only have a single non-zero component anyway.  You might as well just deal with the angles themselves in this case.  
The question of whether rotations can be represented such that they obey a vector addition rule is really only interesting once you get out of a single plane because only then will your prescription for forming the vector really amount to anything other than carrying a few unnecessary zeros.  Once you get out of the plane, it is not going to work.
Also, if you allow for rotation around different points but still rotating in the same plane, then you're again in trouble.  That's just not going to add in any sensible way.
