What harm results from riding with a stretched chain? Due to sheer laziness, my maintenance amounts to an annual trip to the LBS.  After moving, my new LBS suggested that I replace my chain immediately, before it destroys the drive train.  What they didn't mention was that it already had, so I needed a new cassette within a week, because the chain slipped under pressure.

The next year, they checked the cassette before recommending a chain replacement, and it was already getting worn, so I decided to stay with the current chain at least as long as it takes to pay for a new cassette in the chains I'm not purchasing every thousand miles.  At about three thousand miles a year, it shouldn't take all that long.

Now, though, I'm concerned that I might be causing more problems than just cassette wear.  Might I have to replace the chain ring, too?  Am I setting myself up for other potentially expensive problems?  Is the chain likely to break in the middle of a ride within the next year or two?
This is always a controversial topic, with some people arguing on both sides, but in my opinion you should replace your chain when it reaches the official "worn" state (as indicated by a chain stretch gauge).  If you let the chain go the sprockets develop a "hook" and will begin "sucking" the chain.  In addition, shifting performance will suffer.

If a chain is replaced when it first reaches "worn" status, without letting it go too long, the sprockets will not have taken a "set" to the stretched chain and reasonably good drivetrain performance will be retained.

Failing to replace a worn chain results in more cluster wear, more chainring wear, and an increased risk of both shifting difficulty and thrown/broken chains.

You will of course eventually need to replace your cluster and chainrings.  My experience is that you need a new cluster about every five chains, and new chainrings about every two clusters.  Generally when cluster/chainring wear reaches a critical point things go to pot pretty rapidly in terms of shifting problems, slipping, etc.