BOXinfinity is a box containing a bunch of other boxes, including The Final Box and The Other Box.
Some of the more important boxes, which are also of reasonable size, are kept here.
A couple of the boxes located inside of the BOXinfinity are similiar to The Box itself, except being the 1-finality equivalents for different existence ranks. Tthey are, just like the Pseudo-Omniverses in the Allomniverse, smaller and less inhabited, but still significant enough and usefull to earn a spot in this box.
A few of the other boxes are just for structural value, for example, The Hilbert Box is an infinitely dimensional box containing a Hilbert curve that fills space in every natural dimension. The Hilbert curve is of finite order, and the curve houses a ton of civilizations both inside and outside.