What is the highest speed time dilation has been tested?
How close to the Special Relativity prediction did it get?
Answer
Special relativistic time dilation correctly predicts the decay rates of high energy particles. For instance atmospheric muons survive much longer than you would predict based on their lifetime unless you take into a $\gamma$ factor of about 40, for a velocity of $v\approx 0.9997 c$.
The Large Hadron Collider routinely (until the maintenance shutdown started this year!) accelerated protons to an energy of 3.5 TeV (a total collision energy of 7 TeV), which is $\gamma\approx3500$, a velocity $v\approx0.99999995c$.
At even higher energies there is the GZK limit, being tested but likely true, which involves protons moving at a $\gamma$ factor of the order of $10^{10}$! This corresponds to a velocity of $0.999999999999999999995 c$!
EDIT: The opposite limit gives you an idea of how good clocks are these days. It turns out that time dilation has been measured in agreement with special relativity at speeds of 35 km/hr (and also gravitational time dilation at height differences of 33 cm)!!
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