For specifying animations, you should work in the quaternion lie algebra, not in the group as you suggest. There you can represent 1620 degrees without any problem. Furthermore, in the quaternion Lie algebra (pure imaginary quaternions), and only in that space, you can take an arbitrary rotation key, multiply all 3 of its values with 10 and get 10 times that rotation without change in axis.

If you rotate around just one axis, the Lie algebra feels just like Euler angles .. in fact its exactly the same thing, but if you rotate around more than one .. it keeps working intuitively and usably - Euler angles absolutely do not.