Phoo, let us take you at your word. Relativity is a robust theory, it does not depend for its conclusions on the finiteness of real material properties.
So, you have a motor that can supply (if necessary) an arbitrarily large amount of torque. You have a rock with finite rest mass, and you have a rotor with let us suppose arbitrarily large tensile strength.
Obviously as your rotor gets going faster and faster, the inertia of the rock (and rotor) grows, so your motor must use more and more energy for each additional increment of velocity. Additionally, your rotor starts to bend. It may do so because of the treeeeeemendous bending moments you are applying – you didn’t say anything about how stiff your rotor is – but even if not that, it will do so because of the finiteness of the speed of light: your delta V applied at the motor end cannot propagate instantaneously down the rotor to the rock, it can travel no faster than the speed of light, which means the ends of the rotors must necessarily lag the hub.
As you continue to pour energy into your motor, the rotor will continue to accelerate, and continue to bend, eventually into a spiral, perhaps, depending on the length of the rotor arm measured in 1/c and the rate of your acceleration. The inertial mass will continue to climb, and the amount of acceleration you get per unit energy will decline.
You will have increasing difficulty feeding energy to your motor, since conservation of angular momentum means it will be counter-rotating to your mass, and since its moment arm is much smaller, it will be spinning a lot faster. But maybe you can use lasers and such. Or, better yet, instead of a motor and a rock, connect two rockets to each end of the stick, and feed the rockets with lasers or whatnot. Eventually aiming at your system becomes tricky, because its inertial mass becomes big enough to distort spacetime, that is, its surface gravity becomes very high. The gravitational interaction between your orbiting rockets is also going to complicate things. I guess you can swivel their exhausts so they maintain tension in the rotor against their mutual gravitational attraction. I’m pretty sure that when the Schwarzschild radius of the system shrinks to less than the distance between you and the system, you are going to lose contact with it, and be unable to feed it more energy. That is, at that point, it becomes a black hole and whatever happens inside the event horizon is no longer observable to you.
Can the rotor tip ever exceed the speed of light, as measured by you, an observer at rest with respect to the axis of rotation? Of course not. How does that square with its continued acceleration? Because the acceleration is not constant, even if you supply energy at a constant rate and nothing breaks. The inertial mass of the rotor increases asymptotically as the rotor tip approaches c, so that a given increment of energy gets you less and less acceleration. Basically, the energy goes more and more into increasing the mass of the rotor and not into increasings its kinetic energy. The end point, as I said, if nothing breaks, is that you pour enough energy into a small enough volume to create a small black hole, at which point you lose contact with your creation.
