Here's two TikZ solutions!

And here's the code:
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{matrix}
\usetikzlibrary{decorations.pathreplacing}
\newcommand{\bracketwrap}[2][]{%
\tikz[baseline=0pt] \node[anchor=east,align=left,left delimiter=\{,right delimiter=\},#1] (a) at (0,.5ex) {#2};}
\newcommand{\altbracketwrap}[2][]{%
\tikz[baseline=0pt] {\node[anchor=east,align=left,#1] (a) at (0,.5ex) {#2}; \draw[line width=1pt,decorate,decoration={brace,amplitude=4},line cap=round] (a.north east) -- (a.south east); \draw[line width=1pt,decorate,decoration={brace,amplitude=4},line cap=round] (a.south west) -- (a.north west);}}
\begin{document}
He is being \bracketwrap{a nuisance\\naughty} again
He is being \bracketwrap[inner sep=0pt]{a nuisance\\naughty} again
He is being \bracketwrap[inner sep=10pt]{a nuisance\\naughty} again
He is being \bracketwrap[text=red]{a nuisance\\naughty} again
He is being \bracketwrap[align=center]{a nuisance\\naughty} again
He is being \bracketwrap[align=right]{a nuisance\\naughty} again
He is being \altbracketwrap{a nuisance\\naughty} again
He is being \altbracketwrap[inner sep=0pt]{a nuisance\\naughty} again
He is being \altbracketwrap[inner sep=10pt]{a nuisance\\naughty} again
He is being \altbracketwrap[text=red]{a nuisance\\naughty} again
He is being \altbracketwrap[align=center]{a nuisance\\naughty} again
He is being \altbracketwrap[align=right]{a nuisance\\naughty} again
\end{document}
The matrix
library in TikZ defines a couple of keys left delimiter
and right delimiter
which specify left and right delimiters. These can be applied to almost any node, not just matrices. The first set of examples uses these.
However, I find that sometimes I can see the pieces that the extensible braces are made from (look at the third example of the image above), so the second set actually draws the braces using the decorations.pathreplacing
library. You lose the nice tapering at the ends and middle of the braces, but at least the braces are continuous.