After mentioning Brent Burns, I think it is best to mention Joe Pavelski next. Pavelski is another former Shark looking for his first Stanley Cup at the age of 38 now with the Dallas Stars.
He has played in 176 playoff games going back to 2007 and has been to the Stanley Cup Final twice once with the Sharks (2016) and once with the Stars (2020). Of all the players still looking for their first Cup, Pavelski might be the one I am rooting for the most.
A few other names on the Stars to mention: Ryan Suter, Jamie Benn, Evgenii Dadonov and Luke Glendening all are looking for their first Stanley Cup as well.
Benn, the long time captain of the Dallas Stars, has played over 1000 games in the NHL, including 79 playoff games with Dallas going back to 2014. The farthest Benn has been in the postseason was the Stanley Cup Final in 2020, but now with a much deeper team and some very talented young stars on the rise, this could be his best chance at winning a cup.