Also, which part of an integrated circuit has been affected by Moore's law most?
Is it the memory (caches, embedded memory, etc.)?
Or is it the logic (execution units, control units, etc.)?
Or something else?
Ideally with numbers/percentages.
Pretty much everything.
One thing slightly less affected by Moore's Law, as you put it, is some, but very far from all, aspects of power electronics. But even there, finer projection systems, more accurate etching, higher quality vision systems, they all still make a difference in the qualities and capabilities of the devices.
Moore's Law simply predicted the self-driving cycle of higher accuracy processes, that enable the construction of better machines, which then drive more accurate production.
And as is to be expected on a single chip people don't design something for a 22nm process and then also something for a 340nm process, since the latter would be wasting "high cost real-estate space", whereas it could be smaller when designed for 22nm as well, which means more total devices on the same wafer, which means by comparison cheaper Chips.
There's two more exceptions (that I can readily think of) to the rule, which states that the number of transistors on a given square of silicon will grow at a steady exponential rate for several decades (which have already passed, and true to prediction we now talk more about processing power than number of transistors, as it has become all a bit fuddled in modern processing systems). I'm afraid both exceptions don't concern transistors or processing power, so they pretty much fall outside the law.
The first other exception is on-chip capacitances and resistances. They still shrink, as the barrier layers can become smaller with higher accuracy processing, but to the best of my knowledge they do not shrink at the same rate as the transistors do. They may have even already nearly stopped, because we're getting too close to Quantum Effects in the layer sizes we could make inside pure capacitances anyway.
Another exception to a certain degree are MEMS (Micro Electro-Mechanical Systems) and LEDs (Light Emitting Diodes - I know you know, but if I write out MEMS, you know....), these do use "15 to 30 year old" technology. Sometimes literally when it comes to the LEDs on your cheap Chinese LED string with sub-par colour matching and brightness. But even there Moore's Law has driven the developments of the last years, as more experience develops and more readily available machines with 100nm to 1μm feature size capability cycle on a second hand market, allowing easier and cheaper experimentation and trial and error.
Here, Wikipedia what says about a transistor number count on an integrated circuit. Today FPGA are the most complex integrated circuits with more than 20 billion transistors (combination of memory and logic function).
Processors reach ten billion transistors.
Have a look at the roadmap: International Technology Roadmap for Semiconductors.
Process scaling affects all the digital logic to the same extent. It is almost possible to just do a complete die shrink when going from, say 32nm to 22nm, leaving everything in the same place just smaller. (It does require a small amount of re-tuning, but much less work than from scratch). This is the "tick" of Intel tick-tock.
However, given the reduced die size, there is then the option to make it larger again by adding more of something. Maybe more execution units, maybe more cache, maybe a more sophisticated branch predictor. That is the "tock".
(DRAM is produced by a slightly different process using very similar tools, so it also scales at roughly the same rate).