Nothing resembling Moore’s law style advancement, despite a lot of money to be made.
It is likely exaggerated but the headlines claimed the big Australian battery returned the investment within a couple of months. There’s a LOT of money to be made on energy storage.
Why would fusion reactors be moorable?
As you say, batteries have been around for a long time. The rapid growth/improvement part of the battery curve happened back in the late 1800s/early 1900s.
The lead-acid grid lattice design, still the...errr...gold standard when it comes to the most amount of joules stored per buck, was invented in 1881.
(modern technologies like the various lithium battery chemistries win when it come to storage for a given mass -- thus their use in things like portable devices and cars, but lead-acid still wins when it comes to storage for a given cost).
Li-ion has caught up. If you have $400 to spend on batteries both li-ion [1] and comparable lead acid [2] (deep discharge, long cycle life) cost around 3 Wh/$.
[1] https://www.imrbatteries.com/samsung-29e-18650-2850mah-2-75a...
[2] https://www.powerstream.com/BBep.htm (EP100-12)
We also have lots of different designs to experiment with, and much better computers for running simulations. As the computers improve, fusion progress will speed up, if the funding is available.
* (The triple product of temperature, density, and confinement time is the critical fusion metric; for every fusion fuel there's a triple product above which you get net power.)
Sure, more money means more improvements faster, but at best that can only amplify already-exponential progress. Unless, it leads to even more money? Or, that each improvement scales all factors? Or, that one improvement makes it easier to find subsequent improvements (a kind of positive feedback loop; accelerating returns).
Why should money make fusion have Moores-like growth? Not even silicon has it any more...
Because fusion research is critically underfunded and always has been. Like any project, there is probably a point where we'll hit diminishing returns, but right now we're barely keeping the lights on, much less hitting diminishing returns.
https://commons.wikimedia.org/wiki/File:U.S._historical_fusi...
I'd say the key to the parent's premise, is they said it'd be exponential for quite a few years (rather than indefinitely). That's likely correct. The early improvements would probably leap substantially in regards to the output possible. We saw the same thing in nuclear reactor tech.
The original observation is basically a winner-take-all combined with the fact that transistors scale as the square of the minimum dimension. So, if there is linear improvement in dimension, there is exponential improvement in density.
The reason why Moore's Law continued on for so long was that companies were willing to spend exponentially increasing amounts of money to hit the next technology node because of the winner-take-all nature of the product. Anybody who got to the next node forced everybody else to the next node or wiped them out of business.
This is going to be the same thing with fusion. Linear improvements in the fundamentals translate to exponential improvements (fourth power or better) in the outputs. The first folks to fusion are going to force everyone to fusion or wipe them out of business.
At that point, fusion will go onto a Moore's Law curve and we will see exponential improvements for quite a few years.