How do I use a circle as a math accent (larger than \mathring)?

You can use \accentset from the accents package:

$ \mathring{I}\ \mathring{J}\ \accentset{\circ}{I}\ \accentset{\circ}{J} $

In the end I'm using an even larger circle than in Caramdir's great answer: accents sets the \circ in \scriptscriptstyle; I'm using \scriptstyle. To not affect the line spacing so much, I have the circle lowered and let it stick out a bit of the bounding box of the resulting accented character. (In print it looks nicer than on screen ...)

\documentclass{article}
\usepackage{amsmath,accents}
\newcommand{\interior}[1]{\accentset{\smash{\raisebox{-0.12ex}{$\scriptstyle\circ$}}}{#1}\rule{0pt}{2.3ex}}
\fboxrule0.0001pt \fboxsep0pt
\begin{document}
$\mathring{I}\ \accentset{\circ}{I}\ \interior{I}$ \fbox{$\interior{I}$}
\end{document}