Paying 5 cents per extra uniform draw, chosen upfront

You will be paid according to independent $\text{Unif}(0,1)$ draws. The first draw is free; each additional draw costs $0.05, and you must commit to how many additional draws you are buying before anything is drawn. The amount you receive is the largest value among all the draws. With the best choice, what is your expected net payout?

$\tfrac{13}{20}$. The maximum of $n$ uniforms has mean $\tfrac{n}{n+1}$, so drawing $n$ numbers in total is worth $f(n)=\tfrac{n}{n+1}-\tfrac{n-1}{20}$. The $n$-th draw adds $\tfrac{1}{n(n+1)}$, which exceeds the $\tfrac1{20}$ fee for $n\le3$ and exactly equals it at $n=4$, so $f(3)=f(4)=\tfrac{13}{20}$ is the maximum.