An observer in geosynchronous orbit experiences no gravity so that reference frame is different from the one on the ground, even though they rotate at the same rate.

The fact that such an observer feels no g-forces is irrelevant. Local g-forces are not responsible for gravitational time dilation, difference in gravitational potential is. To illustrate the difference compare the calculated gravitational time dilation for the surface of the Earth to that for Uranus and then compare their respective surface gravity

Using a radius of 6378 km for the Earth's radius and 6e24 kg for its mass, you get a surface gravity of 9.842 m/s^2. Using a radius of 25559 km for the radius and 8.68e25 kg for its mass, you get a surface gravity of 8.867 m/s. using the same numbers to calculate the time dilation at these two surfaces you get 0.9999999993025 for the Earth and 0.9999999974820 for Uranus. The clock on Uranus will run slower even it will experience less gravity.

You're thinking of GPS satellites, which are not geosynchronous. In that case, the satellite moves with respect to the ground so you have to account for special relativity as well as gravity.

No, I choose to use geosynchronous orbit for a particular reason. The formula I gave is the accepted equation for any clock in a circular orbit. Since a geosynchronous orbit is in the same rotational frame as a clock on the surface of the Earth, then if your argument that clocks in the same rotational frame run at the same rate, then the only factor that should cause a difference between the two clocks would be the gravitational time dilation. The math does not bear this out.

All terrestrial observers at the same altitude are in the same reference frame so they experience time (and space) in the same way. It doesn't matter how that reference frame moves with respect to any other, even if it is accelerating.

Clocks at the same altitude (relative to mean sea level) do all run at the same rate, just not for the reason you give.

Each of the clocks in your string occupies a different reference frame. I presume there is no gravity in your scenario so the clocks must have rocket engines to maintain their orbits. It is the acceleration of the rocket engines that distorts time.

The acceleration felt by these clocks have no effect on them, this is the basis of the Clock Postulate. The clock postulate has been tested by putting radio-isotopes in high speed centrifuges and measuring how much they decay. By using different radii and speeds you can test a number of different scenarios. You can expose samples to exactly the same centripetal acceleration/g-force but at different tangential velocities, or traveling at the same speed while experiencing different centripetal accelerations/g-forces. The results were that even at 1000's of gs, the acceleration had no effect on the decay rate. It only depended on the tangential velocity. A clock traveling at 0.866c in circle with a radius of 100 light years would only experience 0.007275 g, but would still run 1/2 as fast as one at the center of the circle. A clock traveling at that same speed at a radius of 1 AU would experience 46038 g, and would again run 1/2 as fast as the one at the center.