What happens to a body if it rotates extremely fast? I am thinking on a object, e.g. ball or planet that starts rotating with increasing speed. Let's assume that his speed get's closer to the speed of light, what happens to this object? There are several forces acting. But I always get caught thinking that it will get heavier and heavier because of the additional energy which is needed to accelerate it. Is that all? Or anything else interesting happens?
 A: Note that "c" means linear velocity, not angular. Then, you would refer to tangential velocity.
If an object rotated at relativistic (tangential) velocity, then each "shell" will have a different space and time compression. By contraction of the tangent lengthes, the perceived length of large circles would be smaller than $2\pi r$, leaning to 0 lenght at equator as the tangential speed approach c.
Appart from that, yes you would need more and more energy to accelerate it, and also the material coherency of the object is unlikely to resist to centrifugal force ! 
A: A phenomenon which has been observed in stars and planets is that when the body rotates faster and faster, the object becomes more and more elongated, thus some points in the object are farther away from the rotation axis, increasing the relative moment of inertia and requiring more torque to accelerate the body by the same amount. 
This effect has also been observed in everyday objects and in everyday devices. For example, the governors on steam engines exhibit this effect. When it spins slowly, the weights are almost all the way in. When it rotates faster, the weights swing outward and are further away from the shaft, requiring more torque to maintain constant angular acceleration. 
