There are many questions on series acceleration already on this site, my question is on the limits of these type of approaches as a whole. Can I, by iteratively using acceleration techniques, make a series that converges as quickly as I want?
If so, is there any limit to this approach? For example, perhaps one could make the convergence arbitrarily fast, but not change the rate in an asymptotic sense?
Philosophically speaking (ha-ha) there is no limit to acceleration. However, without precise restrictions on the method of acceleration, the philosophical answer would be practically useless.
And, when you define your method then -- of course -- the limitations will occur.
Nevertheless, Euler and others would successfully accelerate series. And they would iterate their method to ultimately reach another efficiency level. The crucial moment is about catching a pattern during the consecutive passage from the given stage-series to the next one. Thus, by infinite iteration of simple series like the ones for $\ \log 2\ $ or $\ \frac\pi 4\ $ or $\ \frac{\pi^2} 6,\ $ they were obtaining series converging almost as fast as standard Taylor series.