Below is a closed-form evaluation in terms of simple functions.  Let $y=z/(z-1).$  Then
$${}_3F_2(2,n,n;1,n+1;z)=n(-z)^{-n}\Big((n-1)\big(\log(1-y)+\sum_{k=1}^{n-1}\frac{y^k}{k} \big) + y^n \Big)$$
I proved this by using Pochhammer symbol properties, which gets me to a linear combination of ${}_2F_1.$  Then I used a linear transformation to get from the argument $z$ to $y.$  That series can be manipulated to give the logarithm and a finite sum.  Variable $y$ is always negative, but if it is small, the sum will converge rapidly. If you put this on a computer, watch out for large negative $y,$ which occurs for $z$ close to 1.  In the finite sum, I'd probably add terms pairwise. If the $z\sim 1$ case is your most important case, then it's probably worth thinking about this some more.