How fast are cyclotrons? I know this question is kind of vague, but I just want have a handle on how fast a average cyclotron can accelerate particles, and what kind of limits there are...
 A: The cyclotron depends on the fact that the angular frequency is a constant given by $\omega={qB\over mc}$.  However, that equation is in the non-relativistic limit.
The correct relativistic equation is $\omega={qB\over mc\gamma}$, so $\omega$ is not a constant when the relativistic parameter gamma increases from its nonrelativistic value of 1.  $\gamma$ is related to the energy of the particle be accelerated by the equation $\gamma={T+mc^2\over mc^2}$.  This means that the cyclotron will stop working when the kinetic energy T becomes too large.  That is, the cyclotron requires that ${T\over mc^2}\ll1$.    
A: There is a hard limit on an ordinary cyclotron in that they stop working when the particles become relativistic. This limit can be removed with synchrocyclotrons (which change the accelerating frequency as the particles become relativistic), and isochronous cyclotrons (which have a larger magnetic field at a larger radius to account for the relativistic effects).
There is a more important practical limit, however. The output energy of a cyclotron scales linearly with the area of the cyclotron, which makes it unreasonably expensive to go to high energies. Because manufacturing larger and larger D's for the cyclotron is more difficult, the cost scales even faster than that, so the cost scales more than linearly with energy. It becomes unreasonably expensive very, very quickly.
A: In the spirit of a vague question, a typical cyclotron will accelerate a particle from injection energy to extraction energy in 100 to 1000 turns. Depends of the details, of course, and that's just a ballpark figure.
To get from turns to seconds you need the cyclotron frequency, which is also the RF frequency up to an integer harmonic number, but we're usually talking MegaHertz. 
The limitations are that the particles can't be accelerated to relativistic energies, that you need a large and/or strong (but either way that means expensive) magnet, and you need to focus the particles to stop them drifting transversely and hitting the walls.
A: Beyond relativity (which is overcome by making the cyclotron isochronous) and cost issues, there is a fundamental limitation which comes from the extraction area. There, the beam at the energy that you desire and the beam at the previous turn, need to be spatially separated in order to be able to extract only the former. There are some tricks that can be played by introducing orbit bumps and adjusting the tune of the machine, but ultimately the lack of spatial separation will totally impede the extraction.
This is why superconducting cyclotrons will never EVER get as large as their energy-record-holding normal-conducting counterparts, which would in principle not be so crazy in the LHC and ITER era.
