The Menger sponge is quite difficult to print on a standard filament-based 3D printer; it's full of overhangs. Internal support won't work because it can't be removed, which leaves nice Menger sponge prints in the domain of other 3D printing technologies.

On the other hand, the shape prints just great on the diagonal, as a few people have noted. That means it would be possible to print a sponge on the diagonal and glue the halves together, but that seems like cheating.

If you could print the sponge standing on a corner that would be fine, but it would certainly fall over within a few layers. I tried a couple of times to use automatically generated support from different software packages, but they all insisted on trying to support the interior holes that don't need it. Then I was inspired by another design that allows you to print a seemingly unprintable object, a Moebius strip, by using external support that came as part of the original model and was broken off after printing. This design is the result, and so far it seems to work pretty well.

The print doesn't need any automatically generated support; the hollow cone at the bottom holds it well enough to keep it upright during the print. Afterwards you can leave it in place as a display stand, or break each section off by hand. I needed to do a little additional cleanup with a knife to remove the last nub of plastic on the bottom and trim the parts that had become whitened by breaking them off.

As with the other second-order Menger sponge model I made recently, this one needed to be cleaned up by Netfabb Cloud after OpenSCAD rendered it; the STL file included in the download is the fixed version.

I also found that Slic3r's default solid infill setting caused the very tip of the cube to be overfilled and a little wobbly, easily cleaned up afterwards but I'll be testing with a lower setting to see if that helps.