I recall from a course on classical dynamics that angular velocity is a 3-dimensional vector, and angular velocity can be added and subtracted. From this, my understanding is that an object can "rotate about two axes at once" only in the sense that we can add up the two angular velocity vectors and this defines a perfectly sensible rotation. But the final rotation can also be described as a single vector (the sum of the original two) and is thus just rotation about a single axis. Basically, rotation that cannot be expressed as rotation about a single axis cannot exist.
But in this video of a sphere rotating around two axes, it doesn't seem to do this: I can't see any point that holds still relative to the camera (which would also be still relative to the center of the sphere). I don't think I'm missing such a point, either: The gears sweep over all points on the sphere, and none of the gear teeth are stationary.
Further, the hairy ball theorem seems to imply that there must always be a point at zero velocity, at least instantaneously. But maybe that point of zero velocity is moving around constantly, thanks to some acceleration. Without external forces, parts of the sphere can accelerate thanks to internal forces (that's how it holds its shape while it rotates!), so is it possible that there is some internal force causing such an acceleration?
Or is my understanding of angular velocity incorrect, or is there some hidden force that is implicitly acting on the sphere in the animation?