Wedge Power symbol

Here's a simpler answer:

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\extp}{\@ifnextchar^\@extp{\@extp^{\,}}}
\def\@extp^#1{\mathop{\bigwedge\nolimits^{\!#1}}}
\makeatother

\begin{document}
displaystyle: $\displaystyle\extp^k V \otimes \frac{M}{N} \quad \extp Y$

\bigskip
inline style: $\extp^k V \quad \extp Y$

\bigskip
scriptstyle: $X_{\extp^k V}$

\bigskip
scriptscriptstyle: $X_{X_{\extp^k V}}$
\end{document}

With amsmath the cmex font scales properly in subscripts and superscripts.

If you don't like the default location in inline math, you can use \raisebox{<length>}{} to tweak the vertical position. To get the same behavior in display math use \nolimits:

Code:

\documentclass{article}
\usepackage{graphicx}

\begin{document}
Inline math mode it works as $\bigwedge^{\raisebox{-0.4ex}{\scriptsize $k$}} V$.

In display math:
\[
    \bigwedge\nolimits^k V
\]
\end{document}

The problem here is really that \bigwedge is apparently designed to be used as an abbreviation for "wedge a bunch of things together," much like the \sum and \prod symbols. An exterior power is a completely different animal in terms of spacing. While I don't really know what I'm doing here, I've made an attempt to create a macro \Exterior that gives a command with spacing and appearance appropriate for the exterior product:

\documentclass{article}

\newcommand{\Exterior}{\mathchoice{{\textstyle\bigwedge}}%
    {{\bigwedge}}%
    {{\textstyle\wedge}}%
    {{\scriptstyle\wedge}}}

\begin{document}
displaystyle: 
\[
\Exterior^k V \otimes \frac{M}{N}
\]

inline style: $\Exterior^k V$

\bigskip
scriptstyle: $X_{\Exterior^k V}$

\bigskip
scriptscriptstyle: $X_{X_{\Exterior^k V}}$
\end{document}

Output:

As you can see, it's not perfect, but I think it's closer to what we want for an exterior power. Perhaps someone more knowledgable can tweak the macro or rewrite it completely to produce a better result and/or make the code less of a hack.