The proof of Post's Theorem that I am familiar with assumes you have access to as many variables as you wish in your language. Matiyasevich's Theorem by contrast gives a $\Sigma_n^0$-complete formula of the form $\exists y_1 \forall y_2 \cdots Q y_n Q^* y_{n+1} \phi(x, y_1, \ldots, y_{n+1})$ where $Q$ and $Q*$ are the appropriate ending quantifiers. This requires only $n+1$ variables.
The number of variables required to actually demonstrate $\Sigma_n^0$-completeness of particular language is a function of the particular language being used that we might denote $f_\mathcal{L}(n)$. Note that by "language" here I mean a formal language in first order logic that is interpretable as expressing statements of arithmetic.
Can we construct languages for which $f_\mathcal{L}(n) = n^2$? $n^k$? $2^n$? An Ackermann function? Are any upper bounds known?
Edit: Josh Grochow asked about distinguishing between variables and quantifiers. After some thought, I think that the number of variables can always be made to be one more than the number of unbounded quantifiers so the distinction isn't particularly important.