Why bother changing a chain? I have heard often that you should replace a worn chain, that a chain lasts somewhere between 800-1200 miles, and that if you don't replace the chain on time, you'll need to replace the cassette when you get a new chain.

My question is this: instead of going through 4 or 5 chains between cassette replacement, what is the downside of riding the same chain until the cog needs replacement? And, if you do that, how long would you be able to do so?
The cost of replacing worn parts is one part of the issue with excessive chain stretch, safety is another component. 

Safety

This type of problem will take a while to emerge, but if you are going to run the drivetrain into the ground, at some point your safety may be compromised. Excessively worn chains may break, or skip over excessively worn gears.  At some point you can wear the gears so much you can shear the teeth right off.

If this happens when riding out of the saddle (e.g., accelerating hard or climbing) you can actually be thrown off the bike and into traffic, crash and hit your head, crash your sensitive bits into the stem, or otherwise be embarrassed.  The list goes on.

Will this actually happen to you? Who knows, but you should know that it is a risk.

Total Cost of Ownership

If you let the chain wear excessively you will at some point have to replace:


Chain;
Cassette;
Chain Rings; and
Derailleur pulley wheels.


Chain rings alone can be the cost of 2-3 chains depending on the quality of rings you go for (more if you have a triple chainring).  Cassettes usually cost the price of about 2 chains. Let's put the price of derailleur pulleys at 1 chain (this is up for debate). And a chain costs the price of 1 chain.

That brings the total cost of a drive train to 7 chains.  You would therefore need to get about 6000-8400 miles out of your single chain before the economics start to be in your favour.  Do you think you can do it or will the chain break and become a mangled unusable mess?  When that happens you are stuck replacing everything. Even if you hit 6000-8400 mark, how far beyond it will you get?  Only after 8400 miles are you starting to save.

Also don't forget that a really worn drivetrains shift horribly so you will also need to factor that into the cost somehow.  Is a louse experience worth saving a few dollars?  Only you can answer that question.

The only place where this strategy might work is for single speeds as cogs and rings are relatively cheaper and shifting is a non-issue.