What do you mean by "other axes of rotation"? As long as the object is rigid and not acted upon by external forces, its axis should never change, since both the direction and magnitude of angular momentum are conserved.
Wikipedia talks of "chaotic rotation" of astronomical objects, but only over long timescales due to gravitational interactions and thermal effects. On short timescales, its axis shouldn't change much at all, unless you bump into it and apply an off-axis torque.
Alright, that makes more sense, the trick is that the (conserved) angular momentum vector need not be parallel with the angular velocity vector, the simplest example being torque-free precession [0]. It doesn't help that most examples of non-constant angular velocity have external forces in the mix to confuse the reader.
Wikipedia talks of "chaotic rotation" of astronomical objects, but only over long timescales due to gravitational interactions and thermal effects. On short timescales, its axis shouldn't change much at all, unless you bump into it and apply an off-axis torque.