Is there a better bound than $(n^2)^{n^3}$ for the order of the commutator subgroup of a group whose center has index $n?$

Here's an argument for $(n+1)^{n^2}$. I'll try and be a bit formal, at the risk of making it sound mote complicated than it is. Let $c_1,\cdots,c_k$ be an enumeration of the set $\{c_{11},\cdots,c_{nn}\}$. We will write an arbitrary element $g$ of the commutator subgroup in the form $c_1^{p_1}\cdots c_k^{p_k}$ with $0\leq p_1,\cdots,p_k \leq n$.

There exist $p_1,\cdots,p_k\geq 0$ and a sequence $i_1,\cdots,i_m$ such that $g=c_1^{p_1}\cdots c_k^{p_k}c_{i_1}\cdots c_{i_m}$ (for example with $p_1=\cdots=p_k=0$). So we can pick such a representation of $g$ such that $p_1+\cdots+p_k+m$ is minimal, and then fixing that, such that $p_1$ is maximal, and then such that $p_2$ is maximal and so on. Suppose for contradiction that $m>0$. Then we can move the $c_{i_1}$ term back using rule 1 to get a representation of $g$ such that $p_{i_1}$ is larger, which is a contradiction. Suppose for contradiction that $p_i>n$. Then we can use rule 2 and get a contradiction to the minimality of $p_1+\cdots+p_k+m$.

Another small improvement is $(2n+1)^{n(n-1)/2}$ by letting $-n\leq p_i\leq n$ and using $[x,y]=[y,x]^{-1}$.

It's been 7 years since the question was posted, but now I found myself here and I know better bounds, so I'm answering anyway. The simple proof uses the well-known fact that a finite group $G$ is always generated by a subset of $G$ order less or equal than $[log_2|G|]$.

Let $G$ be a group, let $T$ be a (right) transversal of $Z(G)$ in $G$, which has order $n$ by hypothesis, and let $H=\langle T\rangle$. Since clearly $G'=H'$, we can assume that $G$ is generated by $[log_2n]$ elements. Now, by the Reidemeister-Schreier Theorem $G'\cap Z(G)$ can be generated by $n[log_2n]-n+1$ elements and has hence at worst order $n^{n[log_2n]-n+1}$ since it is a finitely generated abelian group of exponent $n$. It follows that $$|G'|\leq n^{n[log_2n]-n+2}$$

Remark: In Multiplicators and groups with finite central factor-groups. Math. Z. 89, 345-347 (1965), James Wiegold provides with even better bounds.