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We know that there are basically 3 fundamental parts of a processor namely :

  1. Control Unit (CU)
  2. Arithmetical Logic Unit (ALU)
  3. Clock

Also we know that they maintain several important and different functions of the processor.

However, if we had to arrange them according to their importance/priority , then which one should have been on the top of the list ?

I mean what is the most critical component of a processor ?

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    $\begingroup$ In any modern cpu, all of them are equally important, I wouldn't consider clock a part of the cpu though. The terms ALU and CU became adopted after von neumann architecture and computers existed before as well. $\endgroup$
    – Rinkesh P
    Nov 16, 2022 at 16:50
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    $\begingroup$ The most important ones are the ones that stop the cpu from working if you take them away. That’s all of them. $\endgroup$
    – gnasher729
    Nov 16, 2022 at 18:15

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The clock is not a significant part of the CPU. Both the controller and ALU are complex. You can add the register bank.

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I would say the control unit.

Every processor must be able to process a sequence of instructions in order.

It's possible (but tricky, and very silly) to design a processor with no ALU**. The processor might have one type of instruction: copy address X to address Y and branch to address Z. I suspect this is enough to implement a bounded Turing machine emulator*, but if not, you could add paged addresses or conditional branches. On this processor, doing any useful computation is rather difficult, but it should be possible. You have to use silly quantities of lookup tables and self-modifying code.

But a processor with only an ALU and no control unit is not a processor because it can't run programs. A program is necessarily a sequence of instructions, so without sequencing, you don't have a program, and sequencing is a control-unit function.

* It will never be unbounded, because you have no way to generate addresses that don't appear somewhere in the program.

** The control unit will have to increment the program counter to read the 3 parts of an instruction, but you can avoid using actual math for this if you add an unused 4th part and use concatenation. If you did choose to add 1 to PC, though, that still wouldn't be an ALU.

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