Is it possible to index an exclamation point?

The characters !, @, and | need to be escaped in an indexing context. For makeindex to understand you mean to mention them rather than use them, you must 'quote' them with a ". The following should do the trick:

\index{"!} % or, in this case: \fun{"!}

I did some playing around, and the problem seems to stem from your \fun command, as I added in an $x$to see if it would print with this, and only prints the x. Also, adding spacing { ! } rather than {!} seems to break the formatting. I left it in so you can see it. Removing the \fun command also removes it from the index altogether, but I guess you probably worked that out yourself.

\documentclass{report}

\usepackage{imakeidx}

%\RequirePackage{imakeidx}

%===----- Functions for indexing and typesetting -----=====
\newcommand{\fun}[1]{\texttt{#1}\index{#1}}
\newcommand{\FUN}[1]{\texttt{#1}\index{#1|textbf}}
\newcommand{\code}[1]{\texttt{#1}}

\makeindex

\begin{document}
%\SweaveOpts{concordance=TRUE}

\chapter{Logical operators}

\begin{table}[!htbp]
\caption{Logical operators.} \label{table:logicops}
\begin{center}
\begin{tabular}{l l}
Operator                & Meaning \\ \hline
\fun{\textgreater{}}    & greater than\\
\fun{\textless{}}       & less than\\
\fun{\&}                & AND\\
\fun{==}                & equal to\\
\fun{\textbar{}}        & OR\\
\fun{\%in\%}            & is an element of\\
\fun{ ! x }                 & NOT\\
\end{tabular}
\end{center}
\end{table}

I hope this helps, I've never personally used an Index before.

I also had other problems with your code when compiling, so I commented everything out except the table. This could be my compiler, so I'll leave that bit to you :)