Diamond needles sound lovely, but not very safe. Dowels are certainly more affordable, but I really didn't like the idea of dowels sticking up out of the base, in case a little person were to fall on it. (I keep remembering the shingle and nail trap on the stairs from Home Alone) Also, the CPSC Choke Test Cylinder defines limits for small toys to prevent choking, so the discs should not be too small, either.
Instead of needles or long dowels, I glued short pieces of dowel halfway in the holes, so the discs can nest on one another:
The base was made from a floor board with three 1/4" blind holes spaced apart enough for the largest discs to fit in them without overlapping. A jigsaw or coping saw would certainly do the trick, but I grew up with a band saw, so that's what I used. First I laid out the design and then roughed in the shape:
I put a scrap piece of shelving on top of the band saw table to support the close work and avoid throwing small pieces:
I made the discs from 3/8" plywood using a set of hole saws like this one:
The puzzle starts to get time-consuming as you add more discs to the set. Five discs don't take long to solve, but the time is approximately doubled every time you add another disc.
If you decide to make this, or think of improvements, please share in the comments.