Unlimited dollar rerolls of a die

Alice rolls a fair six-sided die. After each roll she may stop and collect the number showing, in dollars, or pay $1 to roll again; there is no limit on the number of rerolls. With optimal play, what is her expected net payout?

$4$. The game looks the same after every reroll, so the best strategy is a threshold: stop on $k$ or more. Its value solves $v=\tfrac{k-1}{6}(v-1)+\tfrac16\sum_{j\ge k}j$, which gives $v=4$ for $k=3$ or $k=4$ and less for any other threshold.