\documentclass{standalone}
\usepackage{amsmath}

\makeatletter
\let\old@sup=^% save a copy of original ^ with catcode 7
\catcode`\^=13% make ^ and activate character
\def^#1{\old@sup{#1}}% change definition to call the original ^ with braces

% patch \pr@m@s using `^` with the new catcode.
\def\pr@m@s{% copy of \@pr@m@s code from latex.ltx
\ifx'\@let@token% <-- use U+0027
	\expandafter\pr@@@s%
\else\ifx^\@let@token%
	\expandafter\expandafter\expandafter\pr@@@t%
\else%
	\egroup%
\fi\fi%
}
\makeatother

\newcommand\mesh{\mathrel{\#}}
\newcommand\smallplus{\mathbin{\texttt{+}}}
\newcommand\indep{\mathrel{\perp\!\!\!\perp}}
\newcommand\myrel{\mathrel{\overset{\textsf{a.e.}}{\sim}}}

\begin{document}
\begin{tabular}{lllllll}
$A \mesh B     $ & $A^\mesh     $ & $A^{\mesh}     $ & $A'^\mesh     $ & $A'^{\mesh}     $ & $A''^\mesh     $ & $A''^{\mesh}     $\\
$A \smallplus B$ & $A^\smallplus$ & $A^{\smallplus}$ & $A'^\smallplus$ & $A'^{\smallplus}$ & $A''^\smallplus$ & $A''^{\smallplus}$\\
$A \indep B    $ & $A^\indep    $ & $A^{\indep}    $ & $A'^\indep    $ & $A'^{\indep}    $ & $A''^\indep    $ & $A''^{\indep}    $\\
$A \myrel B    $ & $A^\myrel    $ & $A^{\myrel}    $ & $A'^\myrel    $ & $A'^{\myrel}    $ & $A''^\myrel    $ & $A''^{\myrel}    $\\
\end{tabular}
\end{document}