There are no chores Tuck gives Winnie in Natalie Babbitt's Tuck Everlasting, but he does charge her with a task to fulfill. The task he gives her is keeping the secret of the spring in the wood her family owns.
When Winnie meets Jesse, Miles, and Mae Tuck, she learns that 87 years ago the Tucks passed through the wood and stopped for a drink at the spring. The spring gave them eternal life and froze them in their current ages so that the Tucks never grew older, got sick, or changed in any other way. The Tucks then carry her to their home to speak with Mr. Tuck so that Mr. Tuck can convince her to keep the spring a secret.
While out in the rowboat with Winnie after dinner, Tuck explains to Winnie about the circle of life, how everything is constantly moving and changing and that death is a part of this endless circle, keeping the circle going. Tuck further explains that, since his family members drank from the spring, he and his family members are no longer part of this circle. He asks her to imagine what people would do if they learned of a spring that gave eternal life. Since people naturally hate death, people would "all come running like pigs to slops. They'd trample each other, trying to get some of that water" (Ch. 12). He further asks her to imagine the following:
All the little ones little forever, all the old ones old forever. Can you picture what that means? Forever? (Ch. 12)
Because he sees the threat of people living forever, of even suffering forever, Tuck begs Winnie to keep the powers of the spring a secret.