Using the same figure twice with no new number

With caption it's easy to avoid a caption going in the list of figures.

\documentclass{book}
\usepackage{caption}

\newcommand{\repeatcaption}[2]{%
  \renewcommand{\thefigure}{\ref{#1}}%
  \captionsetup{list=no}%
  \caption{#2 (repeated from page \pageref{#1})}%
  \addtocounter{figure}{-1}% So that next figure after the repeat gets the right number.
}

\newcommand{\fakeimage}{\fbox{Fake image}} % just for the example

\begin{document}

\frontmatter
\listoffigures

\mainmatter

\chapter{A chapter title}

text

\begin{figure}[htp]
\centering
\fakeimage
\caption{A caption}\label{figure:nice}
\end{figure}

\chapter{Another title}

text

\begin{figure}[htp]
\centering
\fakeimage
\caption{This is for the new figure}\label{figure:dull}
\end{figure}

\begin{figure}[htp]
\centering
\fakeimage
\repeatcaption{figure:nice}{A caption}
\end{figure}

\end{document}

It's up to you whether defining a macro holding the figure to be repeated or one for the caption text.

Here's one way, by stuffing the repeated figure content in a saved\vbox. However, in this case, the repeated figure will not be a float. EDITED to resolve List of Figures issue.

\documentclass{article}
\newsavebox\savefigure
\begin{document}
\listoffigures

\begin{figure}[ht]
\centering
\framebox{first figure}
\caption{First caption}
\end{figure}

\savebox\savefigure{\vbox{%
\centering
\framebox{second figure}
}}
\begin{figure}[ht]
\noindent\usebox{\savefigure}
\caption{Second caption}
\end{figure}

\begin{figure}[ht]
\centering
\framebox{third figure}
\caption{Third caption}
\end{figure}

{\centering
\medskip
\noindent\usebox{\savefigure}\par
\bigskip
Figure 2: Second caption\par}
\end{document}