The additional distortion (and power) is actually much less than most people think but it is there, and biased towards the higher frequencies and thus the drivers with the least power handling capability. Look up the frequency spectrum of a square wave and you will see the harmonics actually fall off fairly quickly. IMO to cause any sort of damage you would have to be exceeding your speaker's power capacity routinely. Note that too much power can damage speakers, clipped or not.
Compression I associate with speakers that have exceeded their linear excursion range. A little further and the voice coil will hit mechanical limits, potentially damaging it physically. Overheating can warp the coil and eventually open the wire (preventing further damage but at the trade of no mode sound). Overdriven amplifiers will compress then go into hard clipping, fairly rapidly for SS, as power output rises past its capacity. The power of a square (clipped) waveform is higher than a pure sinusoid, and definitely higher than musical signals.
All this matters little if you are lopping off peaks of only milliseconds, as Wayne said, in the middle of a loud burst of music (or an explosion in a movie -- I defy anyone to distinguish between clipping and a loud blast or gunshot).
Arbitrarily choosing a pair 90 dB/W/m speakers placed 12' away but near a side wall, just 1 W of power will produce about 85 dB at the listening position. That is pretty durn loud. Going up, 10 W will produce about 95 dB, and 100 W about 105 dB, a level I personally find a bit painful.
My less sensitive speakers need about 200 W for the same output, which means an external amp but my Emotiva does just fine. Slightly more efficient speakers, say 93 dB, means you only need 50 W to hit 105 dB. Sensitivity matters, of course, but I am just not sure it is a thing of much concern given the sensitivity ratings of most speakers and the relatively large power output of AVRs and inexpensive amplifiers. Back in the 70's when 25 W/ch was considered average and 100 W/ch was huge, maybe, but not now...
Frankly, I think the bigger problem now is people playing their systems so loudly they are suffering hearing damage without realizing it.
All IME/IMO/etc. - Don (eh?)
"After silence, that which best expresses the inexpressible, is music" - Aldous Huxley