The fastest (by steps/iterations) is Gauss - Legendre algorithm (see AGM method too) and its modifications.
$25$ iterations to get $45$ million correct digits of $\pi$...
But this algorithm is very expansive in time, because each step uses square root calculating (very-very slow operation for huge number of digits).
Release A: (by Brent $-$ Salamin, modification) :
\begin{array}{llll}
a_0 = 1, & b_0 = \frac{1}{\sqrt{2}}, & u_0 = 0, & v_0 = 1;\\
a_{n+1} = \frac{a_n+b_n}{2}, &
b_{n+1} = \sqrt{a_n b_n}, &
u_{n+1} = \frac{u_n+v_n}{2}, &
v_{n+1} = \frac{a_nv_n+b_nu_n}{2b_{n+1}}
\end{array}
and
$$
\pi_n = 2\sqrt{2} \frac{a_n^3}{v_n} \approx \pi.
$$
First steps:
\begin{array}{|l|l|l|r|}\hline
n & p_n & \pi-p_n & \log_{10}(\pi-p_n) \\
\hline0) & \color{gray}{2.828427...} & 3.13165... \times 10^{-1} & -0.50422... \\
1) & \color{gray}{2.95807...} & 1.83515... \times 10^{-1} & -0.73632... \\
2) & \underline{3.14}\color{gray}{066686...} & 9.25783... \times 10^{-4} & -3.03349... \\
3) & \underline{3.1415926}\color{gray}{465191...} & 7.07059...\times 10^{-9} & -8.15054... \\
4) & \underline{3.141592653589793238}\color{gray}{28322...} & 1.79413... \times 10^{-19} & -18.74614... \\
5) & \small{\underline{3.1415926535897932384626433832795028841971}\color{gray}{15228...}} & 5.41713... \times 10^{-41} & -40.26623... \\
6) & ... & 2.39409...\times 10^{-84} & -83.62085...
\end{array}
Each iteration multiplies number of correct digits by $\approx 2$.
Release B: (by Brent $-$ Salamin):
\begin{array}{llll}
a_0 = 1, & b_0 = \frac{1}{\sqrt{2}}, & t_0 = \frac{1}{4};\\
a_{n+1} = \frac{a_n+b_n}{2}, &
b_{n+1} = \sqrt{a_n b_n}, &
t_{n+1} = t_n-2^n(a_n-a_{n+1})^2, &
\end{array}
and
$$
\pi_n = \frac{(a_n+b_n)^2}{4t_n} \approx \pi.
$$
First steps:
\begin{array}{|l|l|l|r|}\hline
n & p_n & \pi-p_n & \log_{10}(\pi-p_n) \\
\hline0) & \color{gray}{2.91421...} & 2.27379... \times 10^{-1} & -0.64324... \\
1) & \underline{3.14}\color{gray}{05792...} & 1.01340... \times 10^{-3} & -2.99421... \\
2) & \underline{3.1415926}\color{gray}{46213...} & 7.37625... \times 10^{-9} & -8.13216... \\
3) & \underline{3.141592653589793238}\color{gray}{27951...} & 1.83130...\times 10^{-19} & -18.73723... \\
4) & \small{\underline{3.1415926535897932384626433832795028841971}\color{gray}{14678...}} & 5.47210... \times 10^{-41} & -40.26184... \\
5) & ... & 2.40612... \times 10^{-84} & -83.61868...
\end{array}
Each iteration multiplies number of correct digits by $\approx 2$.
Release C: (by Borwein brothers) : (see http://www.cecm.sfu.ca/~jborwein/Kanada_50b.html):
\begin{array}{ll}
y_0 = \sqrt{2}-1, & a_0 = 6-4\sqrt{2};\\
y_{n+1} = \frac{1-\sqrt[4]{1-y_n^4}}{1+\sqrt[4]{1-y_n^4}}, &
a_{n+1} = (1+y_{n+1})^4a_n - 2^{2n+3}(1+y_{n+1}+y_{n+1}^2), &
\end{array}
and
$$
\pi_n = \frac{1}{a_n} \approx \pi.
$$
First steps:
\begin{array}{|l|l|l|r|}
\hline
n & p_n & \pi-p_n & \log_{10}(\pi-p_n) \\
\hline
0) & \color{gray}{2.91421...} & 2.27379... \times 10^{-1} & -0.64324... \\
1) & \underline{3.1415926}\color{gray}{46213...} & 7.37625... \times 10^{-9} & -8.13216... \\
2) & \small{\underline{3.1415926535897932384626433832795028841971}\color{gray}{14678...}} & 5.47210... \times 10^{-41} & -40.26184... \\
3) & ... & 2.30858... \times 10^{-171} & -170.63665...
\end{array}
Each iteration multiplies number of correct digits by $\approx 4$ (in fact, it is interlined release B).
Collection of such algorithms: http://www.pi314.net/eng/algo.php.
But, in general, there are no "the fastest" sequence, because if the sequence $\{s_n\}$ converges very fast, then the sequence $\{t_n\}$, where $t_n= s_{2n}$ converges faster.
