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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.