Turns out you can hide the basic avatar by wearing a huge megaprim (the 65K one) as an attachment. This is being used as an alternative to invisiprims that doesn't interfere with the display of alpha textures (or linden trees or water.)
The wise script it to die if it finds itself unattached. ;)
A couple Lindens have said they consider this to be a bug and to not count on it to work in the future. 1
Keep in mind being completely invisible is a serious no-no. It's treated as stalking usually.