A solution with TikZ:
\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\makeatletter
\newcommand*{\circledleqq}{%
\mathrel{%
\mathpalette\@mathcircledtikz{\leqq}%
}%
}
\newcommand*{\@mathcircledtikz}[2]{%
% #1: math style
% #2: math symbol
\tikz[
baseline=(X.base),
inner sep=.5\pgflinewidth,
line width={%
.4pt%
\ifx#1\scriptstyle -.1pt\fi
\ifx#1\scriptscriptstyle -.2pt\fi
},%
]
\node[circle,draw] (X) {$#1#2\m@th$};%
}
\makeatother
\begin{document}
\[
A \circledleqq B_{A \circledleqq B_{A \circledleqq B}}
\]
\end{document}

Remarks:
- The symbol adjusts its size according to the current math style.
- Also the line width decreases with the smaller math styles.
- The vertical position of
\leqq is not changed.
The circle could be a little higher, thus the next example adds some fine tuning:
\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\makeatletter
\newcommand*{\circledleqq}{%
\mathrel{%
\mathpalette{\@mathcircledtikz{1.05\height}{.9\depth}}{\leqq}%
}%
}
\newcommand*{\@mathcircledtikz}[4]{%
% #1: height fine tuning
% #2: depth fine tuning
% #3: math style
% #4: math symbol
\tikz[
baseline=(X.base),
inner sep=.5\pgflinewidth,
line width={%
.4pt%
\ifx#3\scriptstyle -.1pt\fi
\ifx#3\scriptscriptstyle -.2pt\fi
},%
]
\node[circle, draw] (X) {%
\raisebox{0pt}[{#1}][{#2}]{%
$#3#4\m@th$%
}%
};%
}
\makeatother
\begin{document}
\[
A \circledleqq B_{A \circledleqq B_{A \circledleqq B}}
\]
\end{document}

The size of \leqq can also be decreased. The next example uses \textsmaller of package relsize. An alternative is scaling, but this also decreases the line width of the symbol.
\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{relsize}
\makeatletter
\newcommand*{\circledleqq}{%
\mathrel{%
\mathpalette{\@mathcircledtikz{1.05\height}{.9\depth}}{\leqq}%
}%
}
\newcommand*{\@mathcircledtikz}[4]{%
% #1: height fine tuning
% #2: depth fine tuning
% #3: math style
% #4: math symbol
\tikz[
baseline=(X.base),
inner sep=.5\pgflinewidth,
line width={%
.4pt%
\ifx#3\scriptstyle -.1pt\fi
\ifx#3\scriptscriptstyle -.2pt\fi
},%
]
\node[circle, draw] (X) {%
\raisebox{0pt}[{#1}][{#2}]{%
\textsmaller{$#3#4\m@th$}%
}%
};%
}
\makeatother
\begin{document}
\[
A \circledleqq B_{A \circledleqq B_{A \circledleqq B}}
\]
\end{document}

\olessthan. check the documentation (texdoc stmaryrd) not sure whether this will work with marhjax. – barbara beeton Jun 10 '15 at 14:42\mathlarger, etc. are coming from. Otherwise, this looks like a job forstackengine:) – Sean Allred Jun 10 '15 at 14:50\oleqor\oleqqbe used? do you have a published reference that i could present to the unicode committee in a request? – barbara beeton Jun 10 '15 at 17:26@Allredwon't ping me, but@Sean(or@SeanAllred) will. – Sean Allred Jun 10 '15 at 17:31