There are lots of papers on the Internet about sequences with different rates of convergence towards Euler's constant and every year more are published for better rates of convergence. Why is having sequences with better rates of convergence for Euler's constant is important? Is it relevant to proving its irrationality, and if so, how? By reading their papers, I couldn't find the reason why many people spend time on this subject, especially for one specific constant. Any ideas?
An example: https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-018-1670-6
One reason why rate of convergence is studied is because it helps to calculate numbers (such as Euler's constant, in your case) very quickly. This way computers could give many digits of a given number in a short amount of time.
For example, it is known that $$\sum_{k=0}^\infty\frac{4(-1)^k}{2k+1}=\pi$$ Although the summand seems simple, the series converges extremely slowly. But the Chudnovsky algorithm (I don't have enough time to write it down), although looks more complicated, could calculate $\pi$ way more quickly.