About ORAM initalization and running out of space
Are there any scheme to increase ORAM storage with time while maintaining oblivious?


I found this paper that addresses the issue of obliviously resizing:


Moataz, Tarik, Travis Mayberry, Erik-Oliver Blass, and Agnes Hui Chan. "Resizable tree-based oblivious RAM." In Financial Cryptography and Data Security, pp. 147-167. Springer Berlin Heidelberg, 2015.


I also recall a talk by Elaine Shi where she discussed resizable ORAM, but I can't find the relevant paper.