By rotation I'm here only referring to an object rotating in relation to itself, not in relation to any other object. Also I should add that the axis of rotation should be through the center of the object.
For example with a cylindrical object it's easy to see how it can rotate both around the cylindrical axis and around a diameter axis simultaneously, like this:
http://s29.postimg.org/5bdnfjbg3/cyl.gif
But with a sphere I find it harder to visualize several simultaneous rotations.
How many directions can a sphere rotate in simultaneously? How can we visualize different numbers of simultaneous rotations?
Is there a .gif, video, applet or similar that shows several simultaneous rotations of a sphere so that it's easier to visualize it?
This is Euler's rotation theorem. The comment by @mixedmath is already a suitable answer, but it can also be understood as a consequence of one of the more peculiarly-named theorems, the hairy ball theorem. Suppose that your sphere was in fact rotating in multiple directions at once. You can describe what direction each point on the body is moving, and this forms a vector field over the sphere (Imagine combing a coconut in the direction of rotation). At this point the hairy ball theorem says that there is some singular point where there is no defined direction of motion - because it is on the axis of rotation. Since it is a rigid body, everything else must rotate around this axis in circles, and we have simple rotational motion.
(This theorem is actually a bit overkill, since it applies even to nonlinear vector fields, but it makes a good mental image.)
Since OP requested a GIF visualization, consider the following image of a double rotation in 4D, and note that every point on the object is in motion (although the sides of the tessaract are hidden because they would make it hard to see what is going on):
It is this situation (a rotation with no fixed points) that Euler's theorem says is impossible in 3D.