The GIMP doesn't seem to have a Maximum (or a Minimum) filter. Looking at a description of what the Maximum filter does, it seems like it shouldn't be hard to create a Maximum or Minimum filter for The GIMP using Script-Fu.. Two obvious implementations come to mind, one of them O(N*M2) where N is the number of pixels in the image and M is the radius given to the filter. The other obvious choice would be O(N log N), which might actually be a better way to do it.
If your needs aren't too picky, you can also use Filters:Distorts:Value Propogate. Use "More white" for Maximum and "More black" for Minimum and repeat the filter the same number of times that you wanted for "radius". In other words, if you wanted Maximum with radius 6, run the Value Propagate filter with "More white" and repeat it (Ctrl-F) 5 more times
