I was thinking more along the lines of things with cubical symmetry being the basis for building solid objects, because they tile space, and tilings have good symmetry. On the other hand though, icosahedral polytopes can form fractal-like structures like you described in order to capitalize on the symmetry stabilization. They won't be as "solid" as a cubic-based arrangement, but they wouldn't be unstable either (the higher degree of symmetry of the icosahedral polytopes compared to the cube should offset some of the relative instability compared to the cubic tiling).
In 4D, the 120-cell/600-cell would get so much additional stabilization from their high symmetry that they may be comparable or even exceed the tesseractic tiling in stability.