How fast can a wheel spin? A bicycle is going down an infinite hill, is there a limit to how fast the wheels can turn and, hand in hand, how fast the bike can go? The bike cannot break.
 A: Bike can go no faster than speed of light. The rim of the wheel Lorentz – contracts as velocity of bike increases.
The rest length of the rim of the wheel must remain constant. This means that the rim Lorentz contracts, and that the radial extension of the wheel contracts accordingly. The result is that the wheel become infinitely small in the limit that the bike moves with the velocity of light.
If $v$ is velocity on the rim in the rest frame $K$ of the wheel, we have $\Omega=v/R$, where $R=R_0/\gamma$ is the contracted radius of the rotating wheel, and $R_0$ is their radius when they are at rest. The angular velocity of the rotating wheel is then
$\Omega = \gamma v /R_0$
Hence, in this case the angular velocity $\Omega$ must approach an infinitely great value in $K$ when the speed of the bike approaches that of light.
As observed in the hill’s frame $K'$, the distance between the marks on the path each time a point on a rim of the wheel leaves it is
$l'=\gamma 2 \pi R = 2\pi R_0$
and this distance is independent of the speed of the bike, even if the radius of the wheel decreases with increasing velocity, because the distance between the marks depends upon the rest length of the rim of the wheel and not their Lorentz contracted length. Also in this frame the angular velocity of the wheel remains finite even if the wheel have a vanishing radius when the velocity of the bike approaches that of light,
$\Omega'=\gamma^{-1} \Omega = v/R_0$
and hence $\lim\limits_{v \to c} \Omega' =c/R_0$, which is finite.
Details in the "Relativistic Trolley Paradox" in the Am. J. Phys, June 2016. There are two resolutions with either Lorentz contracted radius of the wheel or with constant one.
A: The limit in any real situation (other than the speed of light) would be the tensile stress in the wheel.  Due to centrifugal force the spinning wheel wants to fly apart and is only held together by the strength of the rim.
stress = density * radius^2 * angular velocity^2
So a typical bike wheel with 700mm diameter (0.35m radius) and assumign it was made of pure graphene (the strongest natural material?) with an ultimate tensile stress of 130GPa.  Don't know the density of graphene, so will assume = 1
angular velocity = sqrt( 130GPa / 0.35m^2 ) = sqrt(1E12) = 1E6 rad/s = 165,000 rev/s
With a circumference of 2m this is 165,000*2 = 300,000 m/s  or 0.1% the speed of light
(edit: sorry on a screen will latex it later)
