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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). 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

my guess:

mandelbulb: ~2.7
mandelbox ~2.6

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