How do you measure the mass of the electron very precisely? This week I showed my high school students that if you add the mass of a proton and the mass of an electron the result is higher than the mass of a hydrogen atom, because of the binding energy being negative. I used very precise measurements of the masses, and some student asked how these measurement are actually done.
I did some research and found out how you can do it with Penning traps, but in the case of an electron you actually use a carbon ion with only one electron, as it is described here :
https://www.mpg.de/7961020/electron-mass
It seems to me that if you use an electron bound to a nucleus, there is binding energy involved. If you get rid of it by quantum physics calculations, then you cannot use this result to show that adding the masses of a proton and an electron gives a mass lower than the mass of an hydrogen atom. It would be a circular argument.
Is there another way to determine the mass of an electron very precisely (not the way Millikan did), without any binding energy involved ?
 A: The experiment you mentioned measures the electron mass in an indirect way: It does not measure the mass of the electron, but instead the magnetic moment (see footnote). This is a property that can be predicted by theory (QED) very precisely. The biggest uncertainty in the predicted magnetic moment comes from the uncertainty in the electron's mass (more precisely: the mass of the unbound electron). So the experimenters turned it around: Since the measurement is more precise than the theoretically predicted value, they can make constraints on the electron's mass. They performed additional experiments to test other predictions by QED, in order to be confident that QED works in this regime. I worked on one of the follow-up experiments that did these further QED tests.
The most accurate direct mass-measurement of the electron that I am aware o (it has been a few years!) is a direct cyclotron-frequency comparison between carbon-6+ and a free electron in the same trap, but here you still have to account for the binding-energy of the carbon-ion.
But this is not a problem, because the binding energy can be measured independently through Laser- or X-Ray spectroscopy (or they can be calculated with QED).
Footnote: The measurement is a $g$-factor measurement, similar in spirit to the $g-2$ measurement that has recently been published for the muon. In the muonic case, there seems to be a discrepancy between the QED and measurement. No such discrepancies have yet been found for the electron, but of course, it makes it even more exciting to take a look at the electron again!
A: Millikan's experiment determined just the charge on the electron.
If you want a more accurate determination, there was this question How did scientists manage to measure the charge of electron so precisely?
The specific charge of an electron $\frac{e}{m_e}$ can be found by seeing how much they are deflected by a given magnetic field.  The speed of the electrons is found from the known accelerating voltage.
https://www.niser.ac.in/sps/sites/default/files/basic_page/Specific%20charge%20of%20electron.pdf  (and mass spectrometry)
Then, since the charge is known, the mass of the electron can be deduced.

Depending how accurate you want it, there is also a way here https://en.wikipedia.org/wiki/Electron_rest_mass#Determination
that determines the electron mass from the Rydberg constant.
