Can a CPU (such as the Intel i3/i5/i7/Xeon) with on-chip cache RAM use that as its only functional RAM, without any external memory banks attached?
Or must there be external RAM, and the cache cannot be accessed or used alone?
Modern desktop/server CPUs often have more internal cache RAM than many 1990's computers had in entire system memory, so there should be plenty enough there to run simple code.
CPUs from before cache existed such as the 6502 would be unable to do anything, as the internal CPU RAM only amounted to a few bytes for the address counter and accumulators.
This is not a question of running any sort of modern operating systems, but running simple code programmed into a custom ROM, or hand-entered with a hex input keypad.
See this extremely detailed account of the PC boot sequence: http://www.drdobbs.com/parallel/booting-an-intel-architecture-system-par/232300699?pgno=2
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.