Given: a simple chord…C-E-G…yields perfect harmony.
What if, instead of a chord of 3 notes, we have a harmony of harmonies?
A harmony of C-E-G each in the keys of C, E, and G?
You seem to be talking about a polychord consisting of the C major triad (C-E-G), the E major triad (E-G#-B), and the G major (G-B-D) triad combined. That would give you a chord of the notes:
C-E-G-B-D-G#
(Duplicated notes don’t matter.)
Spell the G-sharp as A-flat and you could parse that as a C major 9 (add flat 13). Basically what you’ve got there is a jazz chord. 
I’m not clear on the question, but yes, that sounds like what it is. Don’t voice them too closely together!
One thing you won’t have, is a perfect harmony.
The reason you wouldn’t have perfect harmony are actually the harmonics. Even that C major chord isn’t perfect in the technical sense. (The most perfect interval is unison, of course, followed by the octave and the fifth.)
If you look at the harmonic scale you will see that the major third is the 5th harmonic, still pretty low in the series, but the three other notes, D, G# and B are numbers 9, 13 and 15. (They’re also odd harmonics, which makes them more dissonant than even ones, generally speaking.)
The harmonics of all those notes played together just create a mess of different frequencies that follows no mathematical principle, which is the basis for consonance.
It should also be mentioned (I almost said noted) that in most modern tunings, you won’t actually get any perfect harmonics, other than the octave. Modern tuning (called equal-tempering) is generally a compromise designed so that all of the harmonics (in any scale) are all very close to perfect, but none of them quite exactly so. So you get a ratio of 1.25992… where you should get 5/4, 1.33484… where you should get 4/3, and 1.49831… where you should get 3/2.
On a piano, not even the octaves are exact. The harmonics of the strings are not perfect octaves. A perfect octave could only be produced with a perfect string.
If you tune a piano with perfect octaves, it will sound sharp, as the harmonics of the strings are slightly flat. Any interval over an octave will sound off, and piano players rarely stay within one octave (unless it’s someone just learning to play Twinkle Twinkle Little Star or something.)
And the reason that such chords are thought of as jazz chords is that the dissonance creates tension. A big part of jazz is managing tension. Two notes a half-step apart, or an octave plus a half step–G and G#–create a huge amount of tension, and are even often called “avoid notes”.
ETA: If you make that an E *minor *triad, then you have built a C major 9, which has lower tension than with the b13. And that is the start of a discussion of how chords are built and extended from the major scale.
Interesting, I did not know that. But that sounds like a technological limitation, not a mathematical one. Even with an ideal waveform generator that got both the fundamentals and the harmonics exact, you still can’t construct a tuning that gives you exact resonances on all of the chords in all scales.
Many synthesizers have workarounds that allow you to re-tune on the fly so all chords sound good. Here is one example from an Emu XL-7 manual:
It doesn’t mean that you can make all the chords sound good with one single setup, though. What these systems allow you to do is tune the instrument so that it’s tuned to mean-tone on C, for example, which means that C, F and G chords will sound good, but distant ones such as B or A flat will sound rather unpleasant.
Fine, if the music you’re playing doesn’t wander far from C major but sticks around closely-related keys, as baroque music generally does. Then, if the next piece you’re playing is in F# minor, you can switch the tuning to mean-tone on F#, but then C major will sound terrible.
And changing within a piece will certainly not sound right, not in the context of classical music, anyway!
You can retune as many times as desired within a piece of music as described in the quote.
What sounds right is a matter of opinion. I think the current fashion for the extreme use of autotune on pop vocals sounds wrong but that doesn’t stop songs with extreme autotune from being number one hits.
What I mean by ‘in the context of classical music’ is using an electronic instrument to simulate the sound of an acoustic one, and that changing temperament in the middle of a Bach fugue would sound as much like a real keyboard instrument as the autotuning in Cher’s Believe sounds like live singing.
If you are not a traditionalist, anything is possible. Here are notes from Wendy Carlos’ Switched-On Bach 2000: