I am interested to know a geometrical structure with maximum surface area and minimum volume. According to me double napped cone may have such property as surface area of a cone is $\pi rl +\pi r^2$ and its volume is $\pi r^2h/3$, where $r$=radius of the base of cone, $l$=lateral height of cone, $h$=height of cone.
You might want to use a labyrinth (taking a 2D one and giving its walls thickness and height to make it 3D). the more complex it is, the more surface you have, while keeping the volume under control (by limiting the walls' thickness and height).
But the ideal would be a fractal : their surface tends to the infinite while their volume tends to zero. these are objects that are infinitely fractioned (they have a "broken" aspect in all scales) examples: Sierpinsky pyramid, menger sponge as user82834 mentioned, fractal trees, Mandelbulb (I'm not sure about the last one though) and many others.
if you want a "regular" geometrical figure, you can limit a fractal to its first iterations, so you won't have any infinities appearing in your calculations.