Is there any way to make lowercase \chi bigger?

You could play around with the regular text adjustments or by using \scalebox{<factor>}{<stuff>} from the graphicx package in the following way:

\documentclass{article}
\usepackage{amsmath}% http://ctan.org/pkg/amsmath
\usepackage{graphicx}% http://ctan.org/pkg/graphicx
\begin{document}
\begin{align*}
  \chi^2_0 =& \tfrac{1}{2} \\
  \mbox{\Large$\chi$}^2_0 =& \tfrac{1}{2} \\
  \scalebox{1.5}{$\chi$}^2_0 =& \tfrac{1}{2}
\end{align*}
\end{document}

amsmath was just used for the align* and \tfrac functionality. If cluttering your input is a problem, you can put this in a macro (here I've chosen the first enlargement style):

\newcommand*{\bigchi}{\mbox{\Large$\chi$}}% big chi

and use it as \bigchi=\frac{1}{2}. You can play around with different font sizes or scaling values.

Exponents are set at the same height for all characters: there's no difference in height between $\chi_0^2$ and $X_0^2$. This is a good thing.

Exponents are set higher when the formula is in display math mode: compare $\chi_0^2$ and \[\chi_0^2\].