How to give the exponent terms in the denominator part of a \frac expression more vertical space?

I think the denominator of your \frac expression suffers from two issues:

• a "cramped" look, i.e., the exponents are not raised much above the baseline
• first- and second-order exponents are raised to (very nearly) the same height

How to remedy these issues?

• To address the first issue, switch to "uncramped" mode -- see the \ufrac macro below.

• To remedy the second issue, I suggest you switch from {n^n}^2 to n^{n^2}. (The two forms are the same, mathematically speaking, but they are quite differnt from a typographic point of view.) That, or use parentheses to visually "enclose" the n^n term.

  Alternatively, consider switching from "standard" \frac notation to inline-fraction notation.

Whatever else you do, please don't use the unnecessary * multiplication symbol. If you can't do without an explicit multiplication symbol, do use \cdot, not *.

\documentclass{article}

%% "uncramped" \frac macro:
%% (Source: @egreg [who else?!], https://tex.stackexchange.com/a/337334/5001)
\makeatletter
\newcommand{\ufrac}[2]{% 
   \frac{#1}{\mathpalette\ufrac@den{#2}}}
\newcommand{\ufrac@den}[2]{#1#2}
\makeatother

\begin{document}
inline math mode, \verb+\frac+ notation

$\frac{n^n}{{2}^{2n} * {n^n}^{2}} % OP's expression
 \quad
 \ufrac{n^n}{2^{2n} n^{n^2}}   
 \quad
 \ufrac{n^n}{2^{2n} (n^n)^2}$ 

\bigskip
display math mode, \verb+\frac+ notation

\smallskip
$\displaystyle 
 \frac{n^n}{2^{2n} n^{n^2}}
 \quad
 \ufrac{n^n}{2^{2n} n^{n^2}} 
 \quad
 \ufrac{n^n}{2^{2n} (n^n)^2}$

\bigskip
inline-frac notation

\smallskip
$n^n/(2^{2n} n^{n^2})           
 \quad
 n^n/(2^{2n} (n^n)^2)$ 
\end{document}

Does this go in the right direction?

\documentclass[fleqn]{article}

\begin{document}
\paragraph{Original:}
$\frac{n^n}{{2}^{2n} * {n^n}^{2}}$
\[\frac{n^n}{{2}^{2n} * {n^n}^{2}}\]

\paragraph{Proposal:}
$\frac{n^n}{{2}^{2n} * {\mathstrut n^n}^{2}}$
\[\frac{n^n}{{2}^{2n} * {\mathstrut n^n}^{2}}\]


\paragraph{With brackets:}
$\frac{n^n}{{2}^{2n} * ( n^n)^{2}}$

\[\frac{n^n}{{2}^{2n} * (n^n)^{2}}\]

\end{document}