Place popcorn in a large bowl, with room for mixing. Fill another bowl with ice water. Butter several plates, one per each person who will be pulling taffy.
Combine sugar, corn syrup, molasses, water, butter, and salt in a large 6-quart saucepan over medium heat; attach a candy thermometer. Bring to a boil, stirring frequently. Tip the pan slightly to melt the sugar crystals on the sides, scraping down with a wooden spoon. Heat until the mixture registers 300 to 310 degrees F (149 to 155 degrees C) on the thermometer or until a small amount of syrup drizzled into cold water turns hard, 7 to 10 minutes.
Pour about 1 cup syrup onto each buttered plate. Let cool until safe to handle like a hot potato, 3 to 5 minutes. Pull syrup between your hands until a light golden brown color is achieved; the longer you pull it, the lighter it becomes. Twist into a long rope about 3/4-inch thick and place on back on a buttered plate to let taffy harden and cool. Break the hardened taffy apart.
Pour the remaining hot syrup carefully over popcorn, while someone else quickly stirs the batch. Butter your hands and dip them in the ice water; working quickly to avoid burns and before taffy hardens, grab some of the popcorn mixture and form into a 3-inch ball. Repeat with the remaining popcorn.