[thinkling]

I believe you only have this problem if your bike has gears based on a set of sprockets (different size chainwheels and a derailleur to move the chain from one to the other).

In Europe (Holland in particular), people ride single gear city bikes (or internally geared hubs) for decades without replacing chains or cogs. When you only have one chainring and one cog, they wear along with the chain, and it takes a very very long time to encounter problems.

It's when you have the sprocket cluster with multiple cogs that are not all wearing equally, that you get problems. Or often the problem on geared bikes doesn't appear until you replace your worn chain and the new chain no longer meshes well with the cogs worn to match the old chain.

[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.

[jen729w]

Or in any flat hipster city. Melbourne (now Canberra) representing!

I ride my single speed (not fixed) every day. It’s my most beloved possession.

[mardifoufs]

Wouldn't the chain still need to be quite intricate and pretty hard to make even then? Though now that I think about it, you could probably make it in a way similar to chainmail, since tight tolerances aren't actually that crucial (the results would suck, but only compared to modern bikes). Though the other parts might be just as hard to make and especially to fit together (the bearings, the gearing for the chains...)