In your paragraph, though (if you didn't write it then forgive me for using "you"), you use the analogy of a children's roundabout that can't move faster than the arm pushing it. The experiments that have been done don't use electromagnetic radiation to push charged particles, though. They use a static electric field to accelerate them.
--->   --->
---> O --->
--->   --->
If you use a proportional font that'll look pretty stupid, but it's meant to be a particle in a uniform electric field. I you change the electric field then the change will propagate at the speed of light, but the acceleration of the particle "O" is due to electrostatic attraction/repulsion, not EM waves.
"The point is that electromagnetic radiation itself has a velocity, namely the speed of light in the medium it's travelling in. Since energy can be transferred to an electron, presumably, only when a wave of energy catches up with it, obviously it's impossible for the electron to ever reach the speed of the wave influencing it." That makes this paragraph somewhat incorrect. [I never addressed this point, since the received view is that cyclotrons operate with huge electromagnets, while Leigh seems to be suggesting they are powered by something like a van de Graaff generator-RW]

An interesting experimental result that convincingly proves the time dilation effect is "Measurement of the Relativistic Time Dilation using mu-Mesons" , D.H. Frisch and J.H. Smith, American Journal of Physics, Volume 31, pp. 342-355, 1963. This doesn't prove the "upper speed limit", but you can't predict time dilation without assuming constancy of the speed of light between inertial frames, so I'll quote their results:

Muons are used as a "clock", because they have a sufficiently short decay time and are produced in large numbers. Their decay distribution was measured at the top of Mount Washington, for muons with speeds between 0.9950c and 0.9954c. 568 were detected in one hour. This distribution is then used to predict the number of muons that would survive to reach sea level, 1910 feet lower. That turns out to be 27 - ie if they repeated the experiment at sea level, 27 muons would be detected in one hour. They then repeated the experiment at sea level and detected 412 muons in one hour.
      That's a staggering difference, and relativity is the only theory that seems to account for it. We can convert that count rate directly into time, knowing the muon's halflife.
      Time taken for (568 - 412) muons to decay travelling at 0.9952c = 6.4usec
      - - - - - - - - - " - - - - - - - - - - at rest = 0.7usec
6.4 / 0.7 = 9.1, and the Einstein time dilation relation predicts a factor of 10.1! This is well within experimental error.
      I'm not a mathematician, and I don't pretend to be able to do the derivation of relativistic effects. However, I know that if you assume the speed of light to be constant for all observers, then time dilation is one of the effects that has to occur. For me, this experiment and others that repeat their findings are excellent evidence for time dilation.

For me, this experiment and others that repeat their findings are excellent evidence for time dilation.
** The problem with all this stuff is that there isn't a single part of it which can be assumed to be solidly established. You yourself state you don't even understand the maths, let alone, presumably, other possible explanations for the reduction in numbers of whatever it is that their detectors are detecting. (There's no indication that you know what procedures they use). You appear to be simply repeating what you've been told.

** The problem with all this stuff is that there isn't a single part of it > which can be assumed to be solidly established. You yourself > state you don't even understand the maths, let alone, presumably, > other possible explanations for the reduction in numbers of > whatever it is that their detectors are detecting. (There's no > indication that you know what procedures they use). You appear to > be simply repeating what you've been told.
The number of muons is decreasing because muons decay. After 6.4usec, a sample of 568 muons will decrease to 27. This is the law of radioactive decay in operation, a very well-researched law which I have verified myself in several experiments, beginning in high school. It has a e^-x form. By taking the appropriate constants, determined in many non-relativistic muon experiments, we can see that for a decay of 568 -> 412 muons, 0.7usec has elapsed according to the muon. There are two possible explanations. One, that the decay of muons is proportional to velocity.
      Other experiments, with other "clocks" have shown that the time dilation effect occurs with them too. In fact, although I have no paper reference, there has been an experiment conducted where cesium beam clocks were synchronised on Earth, and then one flown around in a jet aircraft for some hours. The clock which had been travelling lost a small but significant amount of time. GPS satellites also have to account for this effect.
      As to understanding their procedure, this experiment was actually videoed while it was being conducted. I have looked through their procedure, and they appear to have accounted for all error sources. I would repeat the experiment myself, but unfortunately I don't have a suitable mountain or laboratory. We do have scintillation detectors available, however, and my colleague Terry Mullins has used these to establish the decay rate of muons. I'm not repeating what I've been told, I'm saying what I have confirmed to my satisfaction.