Though I am not a CS engineer that doesnt imply that I should keep myself from getting the correct knowledge about the technology we use daily. As Biggie rightly said its an ignorance on our part.
The question what Biggie has asked is in the real sense how deep we can go logically rather than just taking the things we use daily for granted.
I too am not sure of th answer but I will try to answer to the best of what I think should be the case. Please correct me if I am wrong and add to my knowledge. I have not refered to any Computer Engineering book before answering this question so please excuse me if I sound stupid.
The answers:
1. What is the role of RAM in a computer?
The RAM is the temporary memory used to load the OS and other prorgams and applications while he PC functions. It is a dynamic memeory which can read and write data ( does both) as far as the power is ON. And the processor accesses it to receive any particular data required to be processed and it stores temporarily the data that we input when a particular program runs ( the input we give).
The BIOS which boots the PC and searches and loads the OS to the RAM does it from ROM which is a computer chip on which the data is stored permanenetly on it and cant be modified ie nothing can be written into it after the initial software for booting is loaded into it and stays there permanently.
2. Can a computer function without RAM?
Now this the trick part. To the best of what comes to me is that as the OS creates the Virtual Memory on the HDD by what is called creating a Pagefile when the PC runs out of RAM it is clear that the HDD though a permanent memory it is readable as well as writeable memory.
So the programs can be stored there and accessed by the processor from the HDD in the same way it does from the RAM the only difference being the Speed of the two.
So what I think is that if the ROM chips can hold data to instruct the BIOS to load the OS from the HDD to the RAM it can be made possilble to program such chips with instructions to load the OS directly to the Pagefile. And a certain amount of space be already allocated on the HDD as a sort of permanent Page file where the OS can be loaded and reside and hold some other programs as well.
And from there on the OS can load all other apllications in the Virtual Memory itself just as it does normally when the system runs out of RAM.
And the system can work from here but just with a bugging slow speed. But as far a possibility is concerned I think it can work.
And to add to the justification I can cite the examples of many other chip controlled electronic systems and equipments of daily use making using sensors to control various functions by means of chips holding data the small set of instructions necessary for the working and do not have a seperate dedicated RAM as such. Eg. modern refrigerators, many systems in cars using ICs, washing machines, microwaves etc.
So story cut short : We can possibly have a computer without a RAM and have pre porgrammed ROM chips to load the OS to the Virtual Memory from where all other applications can be loaded and accessed from by the Pocessor. Provided the Pagefile is big enough in space ( MBs) to atleast hold the OS and some other applications simultaneously. and the other possibility with this system would be low speed and limitations on available space on HDD.
I can be totally wrong in the concept technically please excuse my ignorance and correct me.