No, there is no base LaTeX macro holding the title of the \chapter. Let's see why by looking at a call to \chapter{<title>}. Here's the definition of \chapter inside book.cls (similar inside report.cls):
\newcommand\chapter{\if@openright\cleardoublepage\else\clearpage\fi
\thispagestyle{plain}%
\global\@topnum\z@
\@afterindentfalse
\secdef\@chapter\@schapter}
Note that \chapter doesn't take any argument. It merely conditions (via \secdef) on whether you called \chapter with/without a star *. If you called it \chapter*, it executes \@schapter, otherwise it executes \@chapter. Assuming you didn't use \chapter*, here's what \@chapter looks like:
\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
\refstepcounter{chapter}%
\typeout{\@chapapp\space\thechapter.}%
\addcontentsline{toc}{chapter}%
{\protect\numberline{\thechapter}#1}%
\else
\addcontentsline{toc}{chapter}{#1}%
\fi
\chaptermark{#1}%
\addtocontents{lof}{\protect\addvspace{10\p@}}%
\addtocontents{lot}{\protect\addvspace{10\p@}}%
\if@twocolumn
\@topnewpage[\@makechapterhead{#2}]%
\else
\@makechapterhead{#2}%
\@afterheading
\fi}
Here argument #1 is the optional argument, while argument #2 is the mandatory argument you supplied as <title>. If you didn't supply the optional argument, this macro \secdef duplicates #2 for #1, effectively calling \chapter[<title>]{<title>}. It's at this point where one can swoop in and grab either #1 or #2 as your \nowtitle using en etoolbox patch:
\usepackage{etoolbox}
\makeatletter
\patchcmd{\@chapter}% <cmd>
{\ifnum}% <search>
{\edef\nowtitle{#1}\ifnum}% <replace>
{}{}% <success><failure>
\makeatother
As you should see from the definition of \@chapter, #1 is used within \chaptermark and will therefore appear in the header, while #2 is used within \@makechapterhead to set the chapter heading. I chose #1 above, but you pick either (or store both).
\chapter* is a little different. A call to \chapter*{<title>} leads to calling \@schapter:
\def\@schapter#1{\if@twocolumn
\@topnewpage[\@makeschapterhead{#1}]%
\else
\@makeschapterhead{#1}%
\@afterheading
\fi}
Here there is no optional argument, only the mandatory {<title>}. It suffices to patch \@schapter as follows:
\usepackage{etoolbox}
\makeatletter
\patchcmd{\@schapter}% <cmd>
{\if@twocolumn}% <search>
{\edef\nowtitle{#1}\if@twocolumn}% <replace>
{}{}% <success><failure>
\makeatother
Note that \chapter*[<other>]{<stuff>} is not allowed.
Here is a complete minimal example:

\documentclass{book}
\usepackage{etoolbox}
\makeatletter
\patchcmd{\@chapter}% <cmd>
{\ifnum}% <search>
{\edef\nowtitle{#1}\ifnum}% <replace>
{}{}% <success><failure>
\patchcmd{\@schapter}% <cmd>
{\if@twocolumn}% <search>
{\edef\nowtitle{#1}\if@twocolumn}% <replace>
{}{}% <success><failure>
\makeatother
\newcommand{\nowtitle}{}
\pagestyle{headings}
\begin{document}
\chapter[Other]{Stuff}
% \chapter*{Stuff}
\verb!\nowtitle{}!: \nowtitle
\end{document}
It would also be possible to achieve the above patch without etoolbox, if needed. Also, a similar method can be used to store the title used for (sub)\sections:
\usepackage{etoolbox}
\makeatletter
\patchcmd{\@sect}% <cmd>
{\ifnum}% <search>
{\edef\nowtitle{#7}\ifnum}% <replace>
{}{}% <success><failure>
\patchcmd{\@ssect}% <cmd>
{\@tempskipa}% <search>
{\edef\nowtitle{#5}\@tempskipa}% <replace>
{}{}% <success><failure>
\makeatother
Argument #7 and #8 respectively point to (sub)\section[<#7>]{<#8>}.
It is often better using packages which provide similar functionality. For example, nameref and titleref.