Two axes for rotational motion I understand that angular momentum is a vector, etc..
But, what really happens when some object, say a ball for example, is set to rotate along two axes? What would the resulting motion look like? 
 A: Angular rotation is a vector so at any given instant any rigid body can only be rotating about one axis. If the body is rotating freely in space with no external forces then angular momentum is conserved. If the object is spherically symmetrical like the ball you suggest as an example, then the angular velocity is in the same direction as the angular momentum and its motion can only be a simple constant rotation about one axis.
For a more complex asymmetric rigid object the moment of inertia is a symmetric matrix with three perpendicular principle axis. If the rotation aligns with one of these axis it will still have a constant angular velocity, but if not then the angular velocity can itself change direction even while the angular momentum remains constant. There are cases where the angular velocity vector processes around the direction of the angular momentum. This makes it look like it has more than one axis of rotation but really it is one axis that is itself rotating. Here is an animation video to show this
http://www.youtube.com/watch?v=s9wiRjUKctU
More complex motion is possible when all three axis are different as seen in this animation
http://www.youtube.com/watch?v=qEWwIV9Z-eA
In this last video a book which has three different principle moments of inertia is used on the space station to demonstrate some of the variety of motion possible. 
http://www.youtube.com/watch?v=GgVpOorcKqc
A: It looks just like rotation around a different axis with a different rotational speed. Specifically, if you set an object to rotate with angular velocity $\vec\omega_1$ and also with angular velocity $\vec\omega_2$, then it's really rotating with angular velocity $\vec\omega_1 + \vec\omega_2$. The direction of the vector $\vec\omega_1 + \vec\omega_2$ is the overall axis of rotation of the object.
Euler's rotation theorem guarantees that any rotation of a rigid object can be expressed as a rotation around a single axis.
All of this applies instantaneously, in the sense that at any given moment, the body is rotating around a single axis. It is possible that the direction of the rotational axis changes over time, though, and this can lead to more complicated motions that may seem as though they can't be described by single-axis rotation.
A: Rotation is geometrically possible only over one axis. This axis can change in time, but at each instant it will be one.
This is geometrical property of 3d space.
Angular momentum axis does not coincide with rotation axis. Generally, rotation axis do precess around angular momentum axis.
Here is the example of rotating body whose angular momentum is absolutely constant, but rotation axis does vary: 
http://www.youtube.com/watch?v=L2o9eBl_Gzw
A: A rigid body can only rotate by one axis and stay rigid. In fact, the only allowed motion is a screw, whereas a rotation about an axis happens simultaneously as a translation along the same axis (called a twist). Their relationship is called the screw pitch. A pure rotation has pitch=0. 
Now if you are asking what if you have a joint that allows two or more rotations (like a universal joint) then the results is that at any instant, there is only one effective rotation axis.
If you have a sequence of three rotations, with rotation matrices $R_1$, $R_2$ and $R_3$ each one about a local axis $\hat{z}_1$, $\hat{z}_2$ and $\hat{z}_3$ the the total angular velocity vector is
$$ \vec{\omega} = \hat{z}_1 \dot{\theta}_1 + R_1 \left(\hat{z}_2 \dot{\theta}_2 + R_2 \left( \hat{z}_3 \dot{\theta}_3 \right) \right ) $$
A: It is absolutely impossible for a body to rotate instantaneously with respect to two different axes (the equations that give the axis of rotation known at several points always have a unique solution!). What actually happens is that when a body rotates, the axis of rotation changes from one instant to another, but at each instant there is only one axis of rotation.
The angular velocity is not an ordinary 3-vector but a pseudovector (or axial vector). The orientation of a body with respect to fixed axes is given by an orthogonal matrix $\mathbf{R}_t$, and the angular velocity can be computed as:
$$\boldsymbol{\Omega}_t = \dot{\mathbf{R}}_t \mathbf{R}_t^T =
\begin{bmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0 & -\omega_x(t)\\
-\omega_y(t) & \omega_x(t) & 0 \end{bmatrix}$$
It is a common practice to define the pseudovector $\boldsymbol{\omega}(t) = \omega_x(t) \boldsymbol{\hat{\imath}} + \omega_y(t) \boldsymbol{\hat{\jmath}} + \omega_z(t) \boldsymbol{\hat{k}}$, which show that for each time $t$ there is a well defined direction $\boldsymbol{\omega}(t)$ for the rotation axis.
