I like Andre Henriques' rephrasing. The Borwein, Bailey, Plouffe series, with $$a_n={1\over16^n}\left({4\over8n+1}-{2\over8n+4}-{1\over8n+5}-{1\over8n+6}\right)$$ would have radius of convergence $r=16$. Bellard gives a more complicated one with $r=1024$. Pschill has one with 21 terms and $r=2^{30}$. If you'll accept $f(1)=1/\pi$, D and G Chudnovsky give $$a_n={12\over\sqrt{640320^3}}(-1)^n{(6n)!\over(n!)^3(3n)!}{13591409+54514013n\over(640320^3)^n}$$ All of these are taken from Chapter 16 of Arndt and Haenel, $\pi$ Unleashed, which gives full bibliographic citations.

I like Andre Henriques' rephrasing. The Borwein, Bailey, Plouffe series, with $$a_n={1\over16^n}\left({4\over8n+1}-{2\over8n+4}-{1\over8n+5}-{1\over8n+6}\right)$$ would have radius of convergence $r=16$. Bellard gives a more complicated one with $r=1024$. Pschill has one with 21 terms and $r=2^{30}$. If you'll accept $f(1)=1/\pi$, D and G Chudnovsky give $$a_n={12\over\sqrt{640320^3}}(-1)^n{(6n)!\over(n!)^3(3n)!}{13591409+54514013n\over(640320^3)^n}$$ All of these are taken from Chapter 16 of Arndt and Haenel, $\pi$ Unleashed, which gives full bibliographic citations.

EDIT: See also http://mathworld.wolfram.com/PiFormulas.html in particular formulas 93-96 where each term gives another 50 digits of $1/\pi$ (which I guess corresponds to $r$ roughly $10^{50}$). Somehow, the series for $1/\pi$ seem to do better than those for $\pi$. I know there are people who would like us to abandon $\pi$ in favor of $2\pi$, but maybe we should really be expressing things in terms of $1/(2\pi)$.