Paid redraws from a hat numbered 1 to 100

A hat holds slips numbered $1$ to $100$. Each round costs $1: AJ draws a slip, records its number and returns it. He may play as many rounds as he wants, and when he stops he receives in dollars the last number recorded. With optimal stopping, what is the game worth?

$1209/14\approx86.36$. Use a threshold $x$: stop at the first draw of at least $x$. The value is $\tfrac{100+x}{2}-\tfrac{100}{101-x}$ (mean of the accepted draws minus the expected number of $1 fees). Over integers this peaks at $x=87$, giving $93.5-\tfrac{100}{14}=\tfrac{1209}{14}$.