Dave Dyer: electron never at rest, so has no meaning
--and would you contend the mass of anything has no meaning? Bah. I'm trying to get estimates of how much meaning. Anyhow, for a "hydrogen atom" the spectrum is determined by stuff like electron mass & charge, and so if you can measure spectrum, then... and so on.
Nobody so far has addressed Warren's actual question, which seems to raise the problem of how well it is ultimately possible to even define such physical constants. A more conveniently macroscopic example is the length of a day. Variations in the earth's orbit limit the accuracy to which the length of individual days can be treated as a constant. Averaging over an entire year (itself subject to smaller variations) permits greater accuracy, but involves redefining the meaning of "day" in a more sophisticated fashion. Digging further down encounters alarming philosophical questions concerning (for example) the definition of measurement, and the universe within which it is legitimate to take an average --- over a sufficiently long period, the "day" is progressively lengthening. Fred Lunnon
--yah; been pretty disappointed with the mathfun responses to my question so far. One could argue the total #electrons in the visible (i.e. that can communicate with us) part of the universe is <10^80, and their total mass cannot be measured more accurately than uncertainty principles using the spatial and/or temporal size of said portion of universe, and hence I guess that means the electron mass has no meaning beyond about 130th decimal place. Also note, the accessible part of the universe will never get any bigger than a certain constant (which we are near reaching already) so this will stay true. The most accurate physics measurements so far that I can think of have about 18 significant figures (atomic clock frequency ratios). Superstring theory allegedly would predict all the physical constants (in Planck units) as infinitely precise real numbers. I actually think this is an important question and deserves more thought. It pertains to the amount of information there is inside physics theories. Including "the final theory."