What is the most precise physical measurement ever performed?

Carrying the fame of being one of the most precisely verified propositions in physics, the ratio of the gravitational to inertial mass was verified to be unity within $1$ in $10^{15}$ by the MICROSCOPE satellite in $2017$. The earlier best precision was $5\times10^{-14}$, obtained by Baessler, et al. in $1999$.

References:

1. https://en.wikipedia.org/wiki/MICROSCOPE_(satellite)
2. https://en.wikipedia.org/wiki/Equivalence_principle#Tests_of_the_weak_equivalence_principle

The magnetic moment of the electron has been measured to a few parts in $10^{13}$. (Source) This provides an exquisite test of quantum electrodynamics, and calculating the relevant Feynman diagrams has been a Herculean effort over decades.

Note that the more precise tests cited in other answers are basically null results: no difference between gravitational and inertial mass; no difference in magnitude of charge between proton and electron; no mass of photon. So I believe the magnetic moment of the electron is the most precise measurement that is non-null and thus “interesting”.

A few more candidates:

• The quantized Hall resistance is a great and surprising example, since it's an emergent property of rather complicated, "dirty" systems. As stated here (2013) one can measure the resistance to one part in $3 \times 10^{10}$. For this reason this effect is now used to define the Ohm.
• Equivalence principle tests using torsion balances reached precisions of about one part in $10^{11}$ in $1964$, see here. Another existing answer gives a more precise result from a more modern experiment. These can be thought of as either verifying the equality of gravitational and inertial mass, or placing bounds on the strength of long-range fifth forces.
• The electrical neutrality of bulk matter follows because the electron and proton have exactly opposite charges. Treating the charge of the electron as given, tests of neutrality effectively measure the charge of the proton, with one experiment achieving a sensitivity of one part in $10^{21}$ in $1973$. Arguments from cosmology can be used to set much larger bounds, though they require more assumptions.