Chains don't really "stretch" -- the insides of the chain wear with use, resulting in parts with more play and an elongation of a length of chain. You measure chain wear (or stretch) by lining the chain against a ruler. Because the links are exactly 1/2 inch long, the measure from rivet to rivet should be on 1/2 inch increments.
You can see in this photo that the top, older chain has about a 1/16" of wear at one foot. The bottom chain is a brand new, unused 10 speed chain. Click on the photo to see large if you want.
It's not necessary to remove the chain to measure it -- just line a ruler up along a straight part of a chain and measure from the middle of a rivet. If you see more than about 1/16th inch of wear along a six inch segment of chain, it's a good idea to replace the chain.
Why should I change the chain?
Chain wear is important mostly for bicycles with rear derailleurs. Excessive wear leads to poor shifting, and worn chains increases wear on your cogs which makes shifting even worse. Consider that a 10 speed cassette costs about $75, and spending $30 to replace the chain doesn't seem too bad.
I'm not an expert, but I think you can get away with a lot more stretch on singlespeeds or bikes with hub gears. First of all, you can use beefier chains. Derailleur shifting performance is not an issue, so worn cog teeth aren't too big of a deal. Please let me know if I'm mistaken about chain wear for singlespeeds, fixed gears and hub geared bikes.
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Anecdotally, I think you're right about chain stretch not being as much of an issue on singlespeeds/fixed gears. I don't have any empirical evidence, of course, and a badly worn chain is going to chew up expensive cogs and chainrings, but I've experienced no real issues on using a tired chain on my fixed for longer than I probably should.
The measuring tape/ruler method you show is VASTLY superior to the chainwear tools sold by Park and Rohloff. A Park tool will show a brand-new, unused Campagnolo 10-speed chain to be worn past the point of salvage...something about the tool's inability to mesh with the rollers or to take them into account. I'm a little fuzzy on the actual mechanics of it.
The other issue I've heard about chain where on single-speed and hub gears is that with a combination of 1-chainring, 1-cog, and 1-chain, you can let the whole thing wear together. The chain may stretch, but, as it does, it will wear the cog and chainring with it, so that everything still meshes together well, up to a point. When you get to the point where your chain is skipping, it necessary to replace all three pieces, but, depending on what you're paying for chainrings/cogs and how much longer you can ride that system, it may be more cost effective over time.