Distortions that the human ear likes. Are there any ?

Intentional artistic distortion (ie many Beatles recordings from Rubber Soul onwards) = good. That's what the recording artists and their teams wanted.

After market distortion (mastering choices done by engineers years later and those defects added by loudspeakers, turntables, amps etc) = bad. You might like them but the original artists, producers, engineers etc might disagree. 

Or they might not, especially if enough money rolls in as a result. Ultimately it all depends upon what you the consumer wants to pay for.

Moving the discussion sideways, I bought an Avid Acutus Dark turntable recently. It came with a "stabiliser"  or puck for a shorter word to sit on the record to apparently enhance the sound (rerduce distortion??)  Not sure of the theory, but I suppose it made sense. My Linn Lp12 does not have one and indeed cannot due  to the weight factor. The Avid puck threads on to the platter to give tight grip on the record, but to change a record you have to unwind the puck each time. Crazy. So I bought at great expense a Stilpoint puck that is a bit heavier but just slots on top. 
I was not amazingly impressed, but one day by mistake I forgot to put the puck on and wondered why the sound was better. Playing around putting on and off I felt the puck was taking out much airiness and musicality so I now have 2 pucks sitting doing nothing.
My point is that reducing one thing (distortion?) also decreases another aspect, and not necessarily favourably. Not sure why this is though.....

I do not think "most" audiophiles like distortion. Indeed, most audiophiles spend a lot of effort and money trying to reduce it, imo.

But there are those who enjoy euphonic distortion. For example, one thread here got in to quite a bust-up when a user insisted his LP dubs to R-2-R were more accurate than the original LPs. He just couldn’t accept that he preferred the tape dubs because of the addition of small amounts of euphonic distortion. (After all, a dub of an LP can’t contain information not on the original disc. It can only add distortion, however slight it may be.)

cleeds
But there are those who enjoy euphonic distortion. For example, one thread here got in to quite a bust-up when a user insisted his LP dubs to R-2-R were more accurate than the original LPs. He just couldn’t accept that he preferred the tape dubs because of the addition of small amounts of euphonic distortion. (After all, a dub of an LP can’t contain information not on the original disc. It can only add distortion, however slight it may be.)

>>>>Oh, I don’t know, why can’t a copy of an LP be better than the original or contain more information? It certainly seems to be true that a copy of a CD can be better than the original and contain more information. Or perhaps a copy of an LP to cassette. I suspect things are quite a bit more complicated than we realize sometimes. Actually I think the term euphonic distortion probably originated with naysayers attempting to claim such and such a thing couldn’t possibly work. I also just explained why euphonic distortion would be buried in a sea of bad distortions, anyway, so even if there was such a thing as euphonic distortion you wouldn’t be able to hear it. The good distortion would be buried in the noise/distortion. You don’t think you’re listening to undistorted sound when you listen to music, do you?

Well I don't think it could Rilly be better. If it were, then the more tape copies you make from the original tape, then the better the result should sound, yes??