This has to be Detroit by a mile. Jersey did win 3 in the last 10 years, but who other than Brodeur was there for all of them? On the Wings we have Draper, Maltby, Yzerman, Lidstrom, Holmstrom & Shanahan who all have 3. Chelios has at least a couple too. Osgood has two as well. Then from the 02 team Datsyuk, Legace & Williams picked up rings. So off the top of my head that's 25 rings at least for the Redwings. I'll bet no other team has 20.
The closest team might actually be Tampa because they have a lot of their 03-04 team still intact.
Scott Stevens and Scott Niedermayer were also there for all 3
Scott Gomez, Brian Rafalski and Patrick Elias were there for 2000 and 2003, as well as Colin White and Jay Pandolfo.
Alex Mogilny and Oliwa were there for 2000
Grant Marshall and Jamie Langenbrunner were there for 2003.