What is the highest speed time dilation has been tested?

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)!!

Perhaps more interesting than the high energy machines for the purposes of testing special relativity is the Continuous Electron Beam Accelerator Facility (CEBAF) at Jefferson lab. Part of the accelerator work on a mixture of beams having energy that range from 50 MeV to more than 4 GeV at the same time, and still impart the same acceleration to each beam. This is only possible because the speed of the beams is so very close to $c$ that the efficiency of the acceleration is insensitve to the beam energy over the entire range.

When the 12 GeV upgrade finishes that range of energies simultaneously present in the beam will be more than doubled.