every time i find a good method, a better one appears and makes the previous one seem ridiculous, like some time ago when i found a good linear convergence method for pi and i suddenly came across gauss-legendre method, that was muuuuuuuch better then the linear one (its quadratic)
and then i found this wikipedia page that got up to NONIC convergence (but not working ಠ╭╮ಠ)
then i found an Hexadecimalic convergence method for pi (i still did not tested and i only found two sites in the whole google that talked about it!!!)
now i dont have any idea if this is the best algorithm that math has to offer, if there is a even better one thats wating to be discovered or if there are infinitely good that get infinitely hard
the formal question is: is there a bound on how good pi algorithms can get? if yes, what?
In the page given by @Matthew Pilling, the formula $$\frac{1}{\pi} = \frac{1}{53360 \sqrt{640320}} \sum_{n=0}^\infty (-1)^n \frac{(6n)!}{(n!)^3(3n)!} \times \frac{13591409 + 545140134n}{640320^{3n}}\tag1$$ due to the Chudnovsky brothers is really spectacular.
Computing the ratio of successive terms, we have $$\left|\frac{a_{n+1}}{a_n}\right|= \frac{1}{151931373056000}-\frac{1}{303862746112000 n}+O\left(\frac{1}{n^2}\right)$$ while, for the Great Ramanujan's formula $$\frac{a_{n+1}}{a_n}=\frac{1}{96059601}-\frac{1}{192119202 n}+O\left(\frac{1}{n^2}\right)$$
Moreover, $(1)$ is alternating. So, we know in advance how many terms have to be added for $k$ exact decimal places. As a first order approximation $$n_k\sim \left\lceil \frac{13}{849} W\left(\frac{4912}{1321}\times 10^{(2 k-27)}\right)\right\rceil$$ where $W(.)$ is Lambert function.
For $k=1000$, $n=70$.