\documentclass{article}
\begin{document}

\makeatletter
Default:\f@family.\f@series.\f@shape.\f@size.

\f@shape\emph{\f@shape\emph{\f@shape}\f@shape}\f@shape.

\f@shape\textit{\f@shape\emph{\f@shape}\f@shape}\f@shape.

\f@shape\emph{\f@shape\textit{\f@shape}\f@shape}\f@shape.

\f@shape\textup{\f@shape\emph{\f@shape}\f@shape}\f@shape.

\f@shape\emph{\f@shape\textup{\f@shape}\f@shape}\f@shape.

\f@shape\textrm{\f@shape\emph{\f@shape}\f@shape}\f@shape.

\f@shape\emph{\f@shape\textrm{\f@shape}\f@shape}\f@shape.

Not \emph{Emphasized} not.

Not \emph{Emphasized \textit{with italic}} not.

Not \emph{emphasized \emph{Doubly Emphasized} still emphasized} not emphasized.

Not \emph{emphasized \emph{Doubly Emphasized \textrm{with upright}} still emphasized} not emphasized.

Not \emph{emphasized \emph{Doubly Emphasized \textrm{with roman}} still emphasized} not emphasized.

\emph{\emph{Doubly Emphasized} emphasized} not emphasized.


Not \emph{Emphasized
  \[ a=b\]
  still emphasized} not emphasized.

Not \emph{Emphasized \emph{Doubly
  \[ a=b\]
  still doubly emphasized} still emphasized} not emphasized.
\end{document}
