"It's hard to even consider because it's such an unknown," Cullen said. "It's all I've known my whole life. I've played hockey since I was two years old. It's all I ever wanted to do. It's all I've ever dreamed of doing and I thank God every day for giving me this opportunity.
"There are so many people that help you along the way. It's just at a time like this you think back on that and you're so thankful and so humble, it's awesome."
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Cullen, 40, has played 19 seasons in the League and has won the Cup three times, twice with the Penguins the past two seasons, and with the Carolina Hurricanes in 2006. He has 56 points (18 goals, 38 assists) in 123 Stanley Cup Playoff games, and 689 points (248 goals, 441 assists) in 1,366 regular-season games.
There were hints that Cullen may be retiring after the 2-0 win against the Nashville Predators in Game 6 at Bridgestone Arena.
Defenseman Ron Hainsey, who received the Cup first from captain Sidney Crosby, said he was lobbying for Cullen to take the trophy before him.
"I was kind of lobbying for [Cullen] if it's his last game, which I think it is, or [Chris] Kunitz, who has four Cups," Hainsey said. "But it's obviously just a tremendous honor."
Cullen, the oldest player in the 2017 Stanley Cup Final, led Penguins forwards in ice time with 19:42, including 4:42 shorthanded to help kill four Predators power plays.