Archimedes Death Ray
This version uses no materials which would have been available to Archimedes (well, except for air :-).
The primary structure consists of three aluminum rings, about seven inches in diameter, salvaged from an ancient 80 MB removable disk platter assembly (the kind that fit in a disk drive just a little smaller than a washing machine :-).
The rings are stacked together with aluminized mylar between the bottom and middle (scavenged from a bag of chips), and clear plastic between the top and middle, sealed up with rubber cement and bolted together with some 6-32 brass machine screws (had to drill and tap the holes, but fortunately I have a drill press out in the garage, so it wasn't too hard). Put it all together and it looks like this:
(You know, with the chip bag, this is also a little like the myth testing ways to start fires from junk you might find in airplane wreckage, and it is a lot easier to inflate a bag than to polish the bottom of a soda can :-).
An additional key piece of the death ray is the small needle glued through a hole in the side of the middle ring, designed to fill up the inside with air:
Air, from a handy dandy hand operated bulb pump, much like the ones used in blood pressure cuffs (but this one actually came from a sporting goods web site to be used to pump up a helmet).
Put it all together and a fully armed death ray looks like:
Tomorrow will tell the tale (if the sun comes out), will it really incenerate anything, or is seven inches too small when combined with all the factors that will probably prevent a really clean focus with this contraption when inflated?
OK, first test in sunlight had some glitches. The main one being that I've got a leak and have to keep pumping to hold a mirror shape, but that makes it really hard to focus on one spot long enough to ignight it. I need some better materials before I try again.
Make it bigger!
Of course, Mythbusters has pretty much beat this particular myth to death with two or three episodes dedicated to different versions, but if they should ever do a "Greatest Myths Revisited" anniversary episode or something, imagine how much fun it would be to scale up :-).
The largest aluminized mylar sheets I've ever managed to find are the emergency survival blankets sold in some sports stores and online. They are aluminized mylar, and the biggest one I've seen in a catalog is 62 inches in the smallest dimension, so probably a little less than 5 feet is the largest mirror that could be made along these lines (unless the crack Mythbusters research staff can find a source for larger sheets). But 5 feet is a lot bigger than 7 inches, so it would probably be a lot more effective :-).
Then, of course, nothing would prevent you from building an array of 5 foot mirrors if you wanted to scale it up even more. An array might work better than a giant mirror anyway as gravity would probably start to get the better of air pressure at some point.