Rather than redefining \emph
in terms of a copy of itself, it's better redefining it from scratch, using the standard definition as model; the standard definition is surprisingly simple:
% latex.ltx, line 3744:
\DeclareTextFontCommand{\emph}{\em}
because it's \em
that does the hard work. So we can simply do
\documentclass[11pt]{article}
\let\emph\relax % there's no \RedeclareTextFontCommand
\DeclareTextFontCommand{\emph}{\bfseries\em}
\begin{document}
Here is \emph{emphasized text with something \emph{emphasized} inside}
\end{document}
However, double emphasis is really too much: italic is sufficiently prominent. The advantage of redefining \emph
in this way is that you can simply comment out those two lines and return to the default.
Why doesn't your idea work? Because with those definitions, when TeX finds
\emph{text}
it changes it into
\bfemph{text}
and this becomes
\textbf{\emph{text}}
that in turn becomes
{<expansion of \textbf>\emph{text}}
…
Sorry, infinite loop: \emph
will restart the machinery.
bfemph
command?bfemph