There is a nice discussion here on LEGO+Duplo+Quatro compatibility. I posted what I found here where a LEGO storage block is the right dimension to be double a Quatro (4x Duplo, 8x normal). However, a 2x2 Quatro will not stack on the storage block stud and the storage block does not have a working underside to stack/attach.
Are there bigger bricks out there that might work with a Quatro? Maybe the Soft bricks? I think the size is off on them and I don't have any to test out. Are there working giant bricks out there that I am clueless on that might be compatible?
I have seen sources on the Samsonite jumbo bricks and they appear to not be compatible. Here.
Is there a mythical "Octo" brick out there somewhere? I don't think it can exist due the to bottom side of the Quatro brick (which does not have the standard single circle like Duplo and normal).