I note that on the loudest volume, the sound from my iPhone to 3.5mm stereo jack is distorted. Above what volume is sound distorted/compressed?
This question might be hard to answer without some rigorous definition of distortion and qualification of what amount of harmonic distortion is permissible - let alone some clue as to what signal is being played.
In general, the headphone jack is desired to distort sound. Whether its muted or amplified, the output isn't what the recorded sound actually is.
Have a look at these threads fr background:
So - you could get an adapter that generates a line-out level signal and control things from another amplifier or you could experiment with signals you choose and find a setting that doesn't sound distorted to your ears. My guess is most people would set the volume a 60% for most commercial pop music and perhaps a little higher for softer music that isn't compressed so aggressively as pop music.
If you're connecting to a device which is expecting a standard line-in audio signal then keep your output volume to 75%-80% maximum as anything over this is going over the 1volt peak to peak signal which consumer audio works on. Your headphones are different as they are not limited in this way and you should be able to send a 100% signal to them without distortion.