Distortions that the human ear likes. Are there any ?

This is right. Say, some audiophiles use single-output tube amplifiers with output triodes and low feedback or without feedback at all. Such devices produce strong second harmonic and these people decide that this is a sample of ideal sound.

When it comes to harmonics (and I realize this thread is about whether or not we might "prefer" any certain kind of distortion), then overall I'd think that the most "realistic" reproduction of harmonics across the board would be the one most immune to criticism. But, then I don't think in terms of arriving at the best "recipe" for harmonics, but in leveling the playing field for for devices that are supposed dish out harmonics somewhere close to their original relationship.

To do that, we need to look at the biggest cause (IMO) of less than ideal harmonic performance and that is the "other" harmonics - line harmonics - or the "bad" harmonics. Cancel out those and the good harmonics come out unscathed, in virtually any system. And that's more of a noise-floor reduction/power-factor correction thing.

We covered all that in class yesterday. You know, the difference between harmonics and harmonic distortion.

I was playing hooky yesterday, but I still would answer the OP by saying that fundamentally speaking, that I don't think that there are distortions that we might prefer, just those that we might think we prefer until we hear it done better more properly and come to prefer it instead.

But then, I can't rule anything out either...the above is just my 2 cents.