automatic size adjustment for nested parentheses

The nath package with \delimgrowth=1 is very close to your preferred style.

\documentclass{article}
\usepackage{amssymb}
\usepackage{nath}
\delimgrowth=1

\begin{document}
  \begin{equation}
     1 - (1-F(x))^n \\
     \Pr(X_{(1)} \le x) \\
     \mathbb{E}[\min\{X_1, X_2\} ] \\
     ( \pi - \arccos (\frac {y}{r}) )
  \end{equation}
\end{document}

Read the nath guide for details, especially the part about incompatibility with amsmath.

Use it this way with a small negative space between two large parentheses:

\documentclass{scrartcl}
\usepackage{amsmath}
\begin{document}
\delimitershortfall=-1pt
\begin{align*}
\left(\pi-\arccos \left(\frac{y}{r}\right)\right)\\
\left(\pi-\arccos \left(\frac{y}{r}\right)\!\right)
\end{align*}
\end{document}

here is an idea how it can be done with a double (( and/or )). It is also possible to use \left((...( and a \right))...). only the number of \left and \right parentheses must correspond to each other

\documentclass{scrartcl}
\usepackage{amsmath}

\delimitershortfall=-1pt
\let\Right\right
\let\Left\left
\makeatletter
\def\right#1{\Right#1\@ifnextchar){\!\right}{}}
\def\left#1{\Left#1\@ifnextchar({\!\left}{}}
\makeatother
\begin{document}
\begin{align*}
\left(\pi-\arccos \left(\frac{y}{r}\right)\right)\\
\left(\pi-\arccos \left(\frac{y}{r}\right))\\
\left((\pi-\arccos \left(\frac{y}{r}\right)))\\
\left(\left(x + \frac12\right)\arccos(\pi)\right)\\
\left((x + \frac12\right)\arccos(\pi)\right)
\end{align*}
\end{document}

You can do this without using external packages, which can simplify things a bit if the solution needs to work e.g. in KaTeX. You can define your own "growing parenthesis" command by using invisibe subscripts around the parenthesis' contents:

\newcommand{\paren}[1]{\mathopen{}\left( {#1}_{{}_{}}\,\negthickspace\right)\mathclose{}}
\newcommand{\bracket}[1]{\mathopen{}\left[ {#1}_{{}_{}}\,\negthickspace\right]\mathclose{}}

Which gets you exactly the look you want when used at the right places (middle column was modified):

\begin{aligned}
&1-(1-F(x))^n&&1-\paren{1-F(x)}^n&&1-\bigl(1-F(x)\bigr)^n\\
&\Pr(X_{(1)}\le x)&&\Pr\left(X_{(1)}\le x\right)&&\Pr\bigl(X_{(1)}\le x\bigr)\\
&\mathbb{E}[\min\{X_1,X_2\}]&&\mathbb{E}\bracket{\min\left\{X_1,X_2\right\}}&&\mathbb{E}\bigl[\min\left\{X_1,X_2\right\}\bigr]\\
&\left(\pi-\arccos(\frac{y}{r})\right)&&\left(\pi-\arccos\left(\frac{y}{r}\right)\right)&&\left(\pi-\arccos \left(\frac{y}{r}\right)\right)
\end{aligned}
\[ \paren{\paren{\paren{\paren{\paren{}}}}} \]
\[ \bracket{\bracket{\bracket{\bracket{\bracket{}}}}} \]