Modern x86 processors have at least 512K of L2 cache. There are applications which would fit entirely into this amount of memory. Can you run these chips with no RAM attached? If so, is there a way to do it that eliminates the writeback timing penalty when the CPU attempts to maintain RAM coherency?
I don't have a specific application in mind, it's just idle curiosity. I'm certain that somewhere there is a niche application where this would be useful though.