Although well-known to some, arch punches are the latest tool I've discovered that I've immediately put into my prototyping toolbox. I'll talk about how I used them, what they're excellent at (and what they're not), and where to get them.
Arch Punches (or leather circle punches, drive punches, hollow punches, etc) are a sharpened circular tube punch mainly used for leatherworking. Handily, they work almost just as well as cutting paper, especially chipboard or illustration board, which is what many boardgame pieces are cut from. Using these you can make near-production-quality tokens / chits in a huge range of sizes.
For example, I'm going to be sending out nice copies of Lunarchitects to reviewers, and since I don't have a factory to custom make me a small-run of the game, I knew I had to do it myself. My scissors skills aren't the greatest, so by combining 60-pt chipboard, full-sheet labels, and arch punches, I was able to make very good-looking double-sided "coins". Alignment is still a bit off, but I'm improving!
Offered sometimes as sets (really not cheap) to individual sizes, if you've got a specific-size of "coin" or chit or something, arch punches turn out to be rather economical. They're usually between $25-$45 and come in 1/16" increments (or 1mm increments) from 3/16" to 7" (4-52mm). C.S. Osborne has a great range and can be purchased individually, but there are lots of other brands as well (Amazon even has a decent selection). A bit of googling will get you most of the way there.
Other tools you'll need are a mallet of some kind (manufacturers recommend non-metal, which I'd echo because the shafts aren't that hard and could mushroom), a mat (manufacturers also have special thick-plastic punch plates, but I've had plenty of luck on a plastic self-healing mat) the punches, and the material to punch.
Once you get the hang of aligning prints to both sides of a chipboard mat (tips in a future blog post), you just align the punch and it's hammer-time. I've read that some people recommend doing it in as few hammer strikes as possible, but I couldn't really ever get through in fewer than eight, and everything looks fine on my side. One thing I would personally recommend would be to put the self-healing mat on a HARD surface (concrete), not a table, otherwise some of the force from the hammer blow will be transmitted to the table and reduce the effectiveness of the punch.
Have you used arch punches in other creative ways? I can imagine making non-circular chits by overlapping punches, and oval-punches using specific oval drive-punches (they're up to about 1/2", so maybe not for actual game-pieces)...
UPDATE: User Murm3l on reddit asked how they work on foamcore. Since I had some on hand I tried it out:
I'd say it worked pretty well, with one negative being it flatted out one edge of the cut circle before slicing through. It was super easy to cut through (one or two hammer blows). If you're looking for a real cheap and easy way to make lots of chits / tokens, this might be it!