Can a CPU function with nothing more than a power supply and a ROM, using only the internal cache as RAM?
See this extremely detailed account of the PC boot sequence: http://www.drdobbs.com/parallel/booting-an-intel-architecture-system-par/232300699?pgno=2


  Since no DRAM is available at this point, code initially operates in a stackless environment. Most modern processors have an internal cache that can be configured as RAM to provide a software stack. Developers must write extremely tight code when using this cache-as-RAM feature because an eviction would be unacceptable to the system at this point in the boot sequence; there is no memory to maintain coherency. That's why processors operate in "No Evict Mode" (NEM) at this point in the boot process, when they are operating on a cache-as-RAM basis. In NEM, a cache-line miss in the processor will not cause an eviction. Developing code with an available software stack is much easier, and initialization code often performs the minimal setup to use a stack even prior to DRAM initialization. 


You can observe this by running a PC without RAM: it will play a series of beeps. The program that plays those is run from the BIOS Flash ROM.

I've also seen this behaviour on some ARM processors. There will be configuration registers inside the SoC that allow you to use the cache as RAM early on in the boot sequence, in order to run a program that finds, enumerates and configures the DRAM.