I will select Detroit Red Wings.
Originally the Cougars from 1926-1930, and the Falcons from 1930-1932, they began as the NHL rehash of the WHL's Victoria Cougars, Winners of the 1925 Stanley Cup FInals (and losers of the 1926 series). They took on the name Red Wings after the Notorious James Norris purchased the franchise. They won 7 stanley cups between 1937 and 1956, and four more in the past 13 years.... none between 1956 and 1996, though.
Detroit has had many illustrious players in their long history, most notable among them Gordie Howe, arguably the greatest player of all-time. Niklas Lidstrom, an outstanding two-way defenseman, and Steve Yzerman, a superlative two- way forward are two others. But not to be forgotten are sixteen other current or future members of the HHOF whose best years were spent in Detroit: Terry Sawchuk, Ted Lindsay, Red Kelly, Sergei Fedorov, Alex Delvecchio, Sid Abel, Pavel Datsyuk, Marcel Pronovost, Ebbie Goodfellow, Henrik Zetterberg, Norm Ullman, Jack Stewart, Syd Howe, Brendan Shanahan, Bill Quackenbush, and Herbie Lewis.
The only seasons in franchise history that one of these all-time greats did not suit up for Detroit, were their first two, the last two, and the nine seasons from 74-75 through 82-83, 13 total seasons in which they earned just one playoff berth. Hey, it's almost like having good players helps ya win!