In the example below the large markers at the axes are drawn over the axes.
\documentclass{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\begin{document}
\begin{tikzpicture}[baseline]
\begin{axis}[xmin=0, xmax=1, ymin=0, ymax=1, clip=true]
\addplot+[mark=*, mark size=3pt]
coordinates {
(0.0, 0)
(0.2, 1)
(0.4, 0.8)
(0.6, 0)
(0.8, 0)
(1.0, 0)
};
\end{axis}
\end{tikzpicture}
\end{document}

Instead I would like the axes to clip the parts of the markers that leave the axis box area. Also, I would prefer the axis and the axis ticks drawn over the marks. It should like this then:

Is this possible?

axis on topto address the OP's final request. – Jake Sep 26 '13 at 16:23set layers. I thought we had a question about that that I could refer you to, but it doesn't seem like it. Could you open a new question that asks specifically about that (it's not really directly related to this question about clipping markers)? That would help other people who come across the same problem. – Jake Sep 26 '13 at 16:42