What do they learn every two years that allows them to keep doubling the processing power of computers?

What happened was that using current technology (be it today, 10 years ago, 20 years ago) as technology approached its theoretical limits - how thin wiring could be made, how densely transistors could be packed, how to dissipate the heat generated by them - research was always going on to push those limits.

It's still going on, in January 2010, Tyndall National Institute in Cork, Ireland announced the first junctionless transistor which will reduce both size and the amount of heat generated. IBM have already tested transistors that can run at 500GHz.

Moore's Law (basically, doubling of the number of transistors on integrated circuits every two years) may only hold true for the next 20 years, but that's using current or foreseeable technology and techniques. Others take the view that as technology advances that the limit of Moore's Law may not be reached for at least 600 years - maybe never.

One limiting factor is cost. I'm sure TNI and IBM would love to build you a processor using 500GHz junctionless transistors - but could you afford it? Maybe in the future we'll all be using them, but at present you'll be pouring money into a bottomless pit.

Here's a simple example, people were building simple, small electric motors long before there was a proper electrical supply to run them - when the only forms of electricity was either static or Volta piles - technology marches on, but some is way before its time.

Edit: it's not the same techniques at all. There's an old commercial that appeared on British TV years ago, I can't even remember what it was for. It showed Rutherford sat at a table with a carving knife chopping bits off a lump of wood. Rutherford split the atom in 1917 but definitely not by using a carving knife - there's also a limit on what current tooling can do as well.

while the gradual improvement process has worked very well indeed for over 40 years now [inventing tools that permit the 'wires' to be made ever smaller], there is good reason to wonder if progress will come to a screeching halt in the fairly near future.

Intel is approaching the theoretical limits of effective insulation from the next 'wire'. At sizes below that point, the electron would quantumly 'tunnel' through the insulating layer and enter the next 'transistor' over -- which would render the device unpredictable and unusable.

thus, millions and even billions are being spent on discovering different methods of 'remembering' and switching the on/off state -- optical computing, different materials, cryogenic temperatures, and on and on.

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Moore's Law [as this is called] is like all gradual improvement processes -- it breaks down when the conditions at the limits violate any of the fundamental laws of universe [physics].

Every mechanical process known to humanity has a similar law -- one that relates the real cost per unit of output to the cumulative experience of all humanity in producing that output. Some of these observed relationships are very fast [Moore's Law -- cost per computing cycle falls by 50% every 18 months] while others are very, very slow [the per square foot, inflation adjusted, cost of stick built housing in North America declines at about 1 percent per year despite that we've been building them for at least 200 years].

The exact same thing happened with the per passenger-mile cost of air travel -- it declined rapidly and predictably until the increasing speed of the aircraft ran into the speed of sound barrier and then got nearly 'stuck'.

Failing to understand that the rate of improvement in different fields is vastly different is one major forecasting error made by amateurs, including many of our politicians. A second type of error occurs when even the very bright indeed fail to realize that their projection N years in the future will violate some natural law and therefore can not happen.

The third major error type is to assume that some desired invention is possible when it has never actually been demonstrated. [the billions we've spent chasing safe fusion power is an example of this]. And a fourth error is to assume that the timing of the desired invention is predictable. [it isn't -- by definition, when something will be invented isn't predictable -- if it was predictable it could only be because you're already invented it and thus the moment is in the past, not in the future.]

It's just the gradual process of incrementally making the transistors smaller.

Also, it's not actually the "doubling of processing power", it's the doubling of transistor density.