How to robustly store bits of text for later use

If you want BODY to be saved then add stuff dynamically, you are likely best using two macros:

\documentclass{article}
\usepackage{environ}
\NewEnviron{testb}{%
  \global\expandafter\let\csname bar\endcsname\BODY
  \expandafter\xdef\csname barplus\endcsname{%
    \expandafter\noexpand\csname bar\endcsname
    \noexpand\bf Hi
  }%
}
\begin{document}

\begin{testb}
  \bfseries
  Hi
\end{testb}
\show\barplus
\end{document}

If you want to avoid using \BODY you could use xparse

\documentclass{article}
\usepackage{xparse}
\NewDocumentEnvironment{testb}{+b}{\expandafter\gdef\csname bar\endcsname{#1}}{}
\begin{document}

\begin{testb}
  \bfseries
  Hi
\end{testb}
\show\bar
\end{document}

With \unexpanded you can avoid worrying about \protected@xdef.

\documentclass{article}
\usepackage{environ}

\NewEnviron{exercise}{%
  \xdef\savedexercises{%
    \unexpanded\expandafter{\savedexercises}%
    \noexpand\begin{printedexercise}%
    \unexpanded\expandafter{\BODY}%
    \noexpand\end{printedexercise}%
  }%
}
\newcommand{\printexercises}{%
  \savedexercises
  \gdef\savedexercises{}%
}
\newcommand{\savedexercises}{}

\newtheorem{printedexercise}{Exercise}

\begin{document}

Here we talk about addition and show that $1+1=2$.

\begin{exercise}
Compute $1+2$
\end{exercise}

Here we talk about integrals.

\begin{exercise}
Compute the following integrals:
\begin{itemize}
\item $\displaystyle\int_0^x e^{-t^2}\,dt$

\item $\displaystyle\int_1^x \frac{e^t}{t}\,dt$, for $t>0$.
\end{itemize}
\end{exercise}

Now we can print the exercises.

\printexercises

\end{document}

I used \newtheorem just for the example.

With xparse released 2019-03-05 or later:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\NewDocumentEnvironment{exercise}{+b}
 {
  \tl_gput_right:Nn \g_loisel_exercises_tl
   {
    \begin{printedexercise}
    #1
    \end{printedexercise}
   }
 }{}

\NewDocumentCommand{\printexercises}{}
 {
  \tl_use:N \g_loisel_exercises_tl
  \tl_gclear:N \g_loisel_exercises_tl
 }

\tl_new:N \g_loisel_exercises_tl

\ExplSyntaxOff

\newtheorem{printedexercise}{Exercise}

\begin{document}

Here we talk about addition and show that $1+1=2$.

\begin{exercise}
Compute $1+2$
\end{exercise}

Here we talk about integrals.

\begin{exercise}
Compute the following integrals:
\begin{itemize}
\item $\displaystyle\int_0^x e^{-t^2}\,dt$

\item $\displaystyle\int_1^x \frac{e^t}{t}\,dt$, for $t>0$.
\end{itemize}
\end{exercise}

Now we can print the exercises.

\printexercises

\end{document}