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This Can't Go On

I avoided the  'productivity' or 'economic value' to focus on something physically tangible. Markets put an objective value on those, but there's no physical laws to help here. 

Generally you'd expect the  marginal value of information to fall as more information is created.   Aristotle's works vs another youtube video or terrabytes of system logs. The information-density-value-efficiency gets lower as you get bigger. Our own hard drive's content is a good additional example: probably <5% of the contents are high value, contrast to when we all had much smaller storage (e.g. 2.5 inch floppy drives). 

That said, this is analogous to diminishing marginal value (or returns) to scale in economic activity. 

Efficiency of any economic system with respect to fundamental resource usage (information, energy) probably is almost certainly declining in scale. Friction adds up. 

This Can't Go On

The calculations for total bits in the systems is correct:  9.82*10^74 bits (earth) and 4×10^106 bits (galaxy).  The bit limit grows quadratically as you expand out the radius and energy-mass content. 
 
12,395 years of 2% growth to achieve the the upper bound of the galaxy's information content given perfect mass-energy efficiency.     

Stepping back, this exercise in extrapolating 2% growth to extremes can be reduced to just the mathematical statement that any exponential growth will exceed a (constant or sub-exponentially growing) limit  in finite time. Yes...  QED.  

Solutions like expanding  this future-humanity's Boltzmann-economy at a 2% radius per year get you to faster-expansion than the speed of light quickly. Quadratic (or similar orders) is probably the best one can do in the long run.  

TLDR:  in bits, exponential growth at a fixed growth rate forever of an economy is impossible (ad absurdum), but something like quadratic growth forever is not impossible.  

This Can't Go On

The argument here by appealing to too few atoms in the galaxy as a limit to economic growth is poor physics. The fundamental  limit to economic value is that of information density.

Most of the galaxy (and universe) is entirely devoid of matter. Visible matter is exceedingly rare.  But not energy. And energy may be assembled into economically valuable patterns, most clearly as information.  E.g. a radio transmission of an opera to listeners does not require consumption of  atoms. 

An upper bound for information density is given by https://en.wikipedia.org/wiki/Bekenstein_bound and it is exceedingly large, so large that there isn't a fundamental limit on the time frames considered here. 

Likewise, appeals to limits to energy emitted from all stars is considerably lower than the total energy content of the galaxy, by tremendous orders of magnitude. 

As this article is looking at the far-term future, then outlandish possibilities (to us) must also be considered;  a very far advanced civilization might consist mostly of a dance of electromagnetic radiation.  Indeed we're clearly on that path with wifi/5G. 

TLDR: economic growth hasn't a fundamental limit per the  arguments here.