What happens to a gyro inside a gyro? What would present for a gyroscope within a larger gyroscope? Any interesting behaviours? Thanks 
 A: In a three-dimensional space, only one rotation is possible. One object cannot rotate independently in two planes, because two two-dimensional planes cannot be independent in less than a four dimensional space. If you rotate an object in one plane (that often is referred to as "around one axis") and then add another rotation in another plane, both rotations just add up and the result is a single rotation in some resulting plane. Therefore two rotations in a gyroscope would end up just adding up to a single resulting rotation. This also relates to conservation of angular momentum.
The situation is different in a higher number of dimensions. For example, in four dimensions, a single object can rotate in two independent planes. So conceptually you could have something like a gyroscope inside another gyroscope there. However, with two simultaneous rotations, no axes of rotation can be defined. So supporting your gyroscopes each in its own plane, but without an axis, might be an engineering challenge.
In six dimensions, you can have either three simultaneous rotations in independent planes without axes or two rotations each with its own axis. There you could have two gyroscopes, one inside the other, rotating independently each around its own axis. 
