The Menger sponge is a fractal object that has any number of interesting mathematical properties and one unexpected and very attractive physical property, which George Hart explored in a video and a page on his website.

There have been a number of Menger sponges on Thingiverse, including one set up to show the diagonal cut that Prof. Hart discusses. I could never get that part to slice correctly, so I made my own in OpenSCAD. That said, my second-order version gave Slic3r some problems as well, so I've also included the STL after being fixed by Netfabb Cloud. For either first or second-order, you'll need two copies to make a full cube.

The half-cubes should slice and print very easily. As noted in the description, Slic3r has some issues with the second-order file; once I processed it through Netfabb Cloud it was fine, so that STL is included as well.

One other Slic3r tip - for some reason a concentric first layer produces a nice solid first layer for the first-order cube, but gaps for the second-order. I found that rectilinear fill for the first layer worked better.

The first-order examples were printed on an OB1.4 using Printrbot PLA and the second-order were printed on a Printrbot Jr with SainSmart PLA, both at 0.3 mm layers, 20% infill, 3 perimeters and top/bottom. The Printrbot Jr lacks a heated bed, so the bottom of the print curled a little bit; it's not noticeable by itself, but keeps the two halves from fitting together well.