What harm results from riding with a stretched chain?

The worst problem I face when waiting too long to replace a chain is that a new chain will skip badly over the worn sprockets.

In my experience, it has a lot to do with the kind of terrain you ride (on- vs. off-road), the amount of rain/mud/snow you take, and if your have full fenders or not.

If you do mountain bike, 500 miles is sometimes enough to need a chain replacement. 1500 miles will for sure damage your sprockets permanently. Now if you ride only asphalt, use full fenders, and take proper care on cleaning and lubing, you might go over 3000 miles without much problem.

I agree with a previous answer, it takes three to five chains to need a cluster replacement if you care well, but the chainrings use to last much longer to me than two clusters. I think it is because, when you install a new chain, the wear pattern on the teeth (both the sprockets and the chainrings) tend to get back to normal (wear occurs at the base of the teeth, instead of their tops). Since the rings wear much less than the sprockets, they go back to normal more easily, and have a much lesser tendency to skip.

It is worth mentioning that I had never experienced a chain braking because it was worn out, so the problem is not dangerous, just expensive (because you have to replace two or three parts instead of one), not to mention the bad ride quality due to worn-out drivetrain.

Not really an answer, but too long for a comment.

I've gone the other way - my second hand road bike has worn/munched big and middle chainrings, and the cassette is pretty worn too. Chain is unstretched with surface corrosion, and the master link popped under load yesterday.

So my plan is to run the entire transmission into the ground, and buy new rings/cassette/chain/jockey wheels all at once. There's no point replacing one thing with new when other parts are approaching terminal.

A road chain is good for ~3000 km / 2000 miles before it starts wearing into the cassette, whereas most chainrings are good for "the life of the bike" or 20,000 km

A worn cassette and worn chain can do more than 3000 km, but they need replacing together. MTBs off road will wear quicker.

I log every ride with strava, so it will be interesting to see how long this setup can go for before shifting and chain jump get too bad.

The wear issue usually ends when the chain starts jumping out of the sprockets (chainring or cassette) while applying heavy loads. This can be dangerous if you are used to dealing with traffic issues. A good chain lube should be used in the early stages of wear . . . it will only make jumping and slipping worse. I have never broken (thrown) a chain unless it was not installed correctly. Avoid connecting links . . . use a chain breaker instead

I know cyclists, who have rode their stretched bike chains beyond the specified limits, and you will be surprised how much mileage, you can still get after this point. Sometimes twice the guide range. I think it depends on what each cyclist - wants to get out from the cycling experience, risk tolerance and budget of cause. All these factors in addition to the specified gauge limit on bike chain change intervals, will help determine (and inform) the frequency of the change.

You need to differentiate two cases:

1. Chain-shift bikes

   In this case, a worn down (=lengthened) chain will wear down the cogs faster, and relatively soon the teeth of the cogs are so diagonal that the chain will start skipping when you apply a lot of force. Once this happens, both the chain and the cassette will need to be replaced. A chain is significantly cheaper than a cassette, so it's worth replacing the chain pretty early to lengthen the life time of the cassette.

2. Single speed and internal gear hubs

   Just as in the first case, a worn down chain will wear down the sprocket. However, you will not get a skipping chain, ever: The teeth of single speed sprockets are significantly longer than those of a cassette, and without a derailleur, it's plain impossible for the chain to skip. Yet, even though the sprocket will retain the chain, the chain will wear round depressions into the flanks of its teeth.

   When your sprocket has this kind of wear, it's a very bad idea to replace the chain without also replacing the sprocket. The form of the teeth will hinder the chain from gliding smoothly into the sprocket, wearing down the chain at an insane rate.

   The ultimate failure mode is that the sprocket starts loosing teeth. You will still be able to ride the bike with a missing tooth, the sprocket will still be able to retain the chain. But once the sprocket looses two consecutive teeth, it's game over, you'll be pushing your bike back home.

   So, you may either replace your chains frequently to avoid wearing down the sprocket, or you may basically wear the set down as far as you like. But in the later case, you must always replace sprocket and chain together. Otherwise the wear on either will destroy your investment in the other.