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It seems like a completion of the process to work out the fractal dimension for these fractals.

It might not be that easy, it seems there are at least 5 variations of fractal dimension (http://en.wikipedia.org/wiki/Fractal_dimension (http://en.wikipedia.org/wiki/Fractal_dimension)). But I think the easiest procedure would be as follows:
Set the max iterations to really high (so it doesn't affect the results)
count the number of voxels with solid threshold at 1e-1, then again using e-2, e-3 etc. The dimension is then ln(number of voxels)*ln(threshold) as threshold approaches 0

For the Mandelbulb you would need to count the number of surface voxels, which you can know because one of 6 neighbouring voxels will be outside the set.
For the Mandelbox you would count every inside voxel.

Both are multi-fractals as the volume of space isn't scaled uniformly in either of the fractals, this means the dimension must be an average fractal dimension.

Anyone want to take a guess at the fractal dimension for each?
My guesses are:
Mandelbulb- 2.6
Mandelbox- 2.8

I think the surface of the Mandelbulb ought to have Hausdorff dimension 3, since the boundary of the Mandelbrot has Hausdorff dimension 2. But you're not describing Hausdorff dimension, are you... that's box counting dimension, I think. The two tend to be similar, though, so I'd say the Mandelbulb should have dimension between 2.8 and 3.

my guess:

mandelbulb: ~2.7
mandelbox ~2.6

i think a simple box counting method would give a good estimation