Cash out the last roll or risk a six

You roll a fair six-sided die. A $6$ ends the game and you receive nothing. Any other value $k$ gives you a choice: stop and collect $\text{\char36}k$, or roll again, with the same rules applying to the new roll. If you play to maximise your expected payout, what is that expected payout?

$\text{\char36}3$. Given a non-six, the roll averages $3$, which is the value of continuing, so you stop on $3$ or more and reroll $1$ or $2$. The value then satisfies $V=\tfrac16(3+4+5)+\tfrac13 V$, so $V=3$ (stopping only on $4$ or $5$ gives the same value, since you are indifferent at $3$).