[Psychlist]

Nope, with single speed/hub gears chain wear is still a problem, it's just that they're less sensitive to it and there's fewer parts to replace. When the chain gets longer it gets loose, but it has to be very loose indeed to fall off or skip on a single speed. With a derailleur setup there's a tension arm with a weak spring so when the chain wants to skip that tension arm lets it. Derailleur setups commonly have some cogs with fewer teeth than single speeds so the problem is more obvious. Also, often derailleur cogs are aluminium while single speed ones are steel (not always!).

Typically a safety bike will get through 3-5 chains before needing to replace the rear cog, and many more before replacing the chainring(s). But Pinion gearboxes in the bottom bracket often run small chainrings that are similar in size to the rear cog, and I suspect they need to replace both rather than just the rear one.

[mrob]

I ride a bicycle with a hub gear as my main form of transport. About two years ago, I noticed the sprocket (rear cog) was getting very worn, so I removed it so I could reverse it and let it wear on the other side of the teeth (standard procedure for this sprocket to extend the wear life). But I accidentally put it back the same way as before and didn't notice until recently when I changed the hub oil. I had no problems with the chain skipping, despite the heavy sprocket wear (although it's now worn to the point that I no longer feel comfortable letting it wear on the other side of the teeth, so I'll have to buy a new sprocket). But you are correct that the chain wears more than the sprocket; I've already replaced the chain a few times.

[Psychlist]

https://ibb.co/0sJx4SN (new cog on the left, old on the right)

I'm not saying you should do this. And Rohloff strongly suggest you don't do this I'm just saying that you can do this.

[askvictor]

The other factor is that single-speed/hub-gear chains are wider, so tend to last a lot longer.