I am trying to prove the additivity propety for the angular velocity vectors acting on a body in three dimensions. Particularly also in the case when the axes of rotation do not cross each other (the body may not be rotating about a fixed point).
In plane motion it is easier to see. Grab two coins, fix one flat on a table and make the other one rotate around the edge of the fixed one without slipping. One can see that the total angle of rotation of the one rotating is 2 times the angle that it has traveled around the fixed one. One can build a proof from this for plane motion (the case when angular velocity vectors are parallel to each other) but i don't know how could one show the same thing for 3d.