A decent frequency counter can resolve a frequency difference of 1E-15 in a few minutes of averaging. Modern GPS-based frequency comparisons (dual-band, post-processed using precise point positioning) are also accurate to the 1E-15 level with around one day of averaging. Note that these counters operate with radio frequencies. The best atomic clocks use optical transitions and are accurate to around 1E-18. In this case, one gains around 6 orders of magnitude in frequency resolution by translating the laser frequency into a radio frequency using an optical frequency comb. The accuracy of the relative frequency measurement is not limiting at all.

The gravitational redshift amounts to around 1E-16 (not 1E-15) when moving the earth one meter closer or further from your clock. You standing next to it is going to have absolutely no effect.

An increase, but those effects are noticed with traditional atomic clocks too. The simple act of moving them between biuldings results in detectable GR effects: time dilation from acceleration as someone pushes the cart. A more accurate measure of such things could yield a functional GR-based accelerometer/gyroscope.

> Mind-boggling stuff.

This and the recent advances in clock precision. In the last few years cryogenic sapphire clocks have achieved short-term stability on the order of 10s of attoseconds.