That would depend on what you consider the "modern era". If you are referring to the post 1967 expansion, then Jacque Lemaire and Yan Cournayer both have 8 cups (All with Montreal between '68-79).
If you are referring the consumption of the WHA (post 1980), the Glenn Anderson and Mark Messier have 6 (5 with Edmonton between '84-90 and 1 with the New York Rangers in 1994).
Modern Era could also refer to the current 30 team format (2001 and later). As of the 2011 Stanley Cup there are 21 Players who have won 2 (7 players from the 2002 and the 2008 cup winning Detroit Red Wings, including Dominik Hasek, Chris Chelios, Darren McCarty, Chris Draper, Pavel Datsyuk, Kirk Maltby and Nick Lindstrom) + Chris Dingman and Nolan Pratt ('01, '04),Oleg Tverdovsky ('03, '06), Scott Niedermayer ('03, '07), Brian Rafalski ('03, '08), Ruslan Fedotenko ('03,09), John Madden ('03, 2010), Cory Stillman ('04, '06), Craig Adams ('06, '09), Andrew Ladd ('06, 2010), Mark Recchi ('06, 2011), Chris Kunitz ('07, '09), Shawn Thorton ('07, 2011) and Tomas Kopecky('08, 2010).
Finally, The modern era might refer to just the post 2004-2005 lockout. In which case 6 players have 2 cups (Adams, Thorton, Ladd, Recchi, Kunitz, Kopecky).
Red Kelly is the most successful NHL player who never played for Montreal. He won 8 Stanley Cups with Detroit and Toronto between 1950 and 1967.
Henri Richard won eleven (11) Stanley Cup championships playing as a member of the Montreal Canadiens.
Ken Dryden won 6 Stanley Cups in a career that only spanned 8 years.
The Montreal Canadiens is the only team in NHL history to have won 5 Stanley Cups in a row. They accomplished that feat between 1956 and 1960 inclusively. They also won 4 straight cups from 1976 to 1979. Other streaks worth noting are the NY Islanders who won 4 cups in a row from 1980 to 1983. The Edmonton Oilers won 4 in 5 years (1984, 1985, 1987, 1988)The Montreal Canadiens from 1956 to 1960, winning 5 Stanley Cups.
The player with the most rings is Henri "the pocket rocket" Richard who has a total of 11 Stanley cup rings , and is also the younger brother of one of the greatest players of all time: Maurice Richard
Henri Richard, with 11
Henri richard with 11 cups
Red Kelly is the most successful NHL player who never played for Montreal. He won 8 Stanley Cups with Detroit and Toronto between 1950 and 1967.
Henri Richard of the Montreal Canadiens leads with 11 Stanley Cups as a player. Jean Beliveau and Yvan Cournoyer also of Montreal, have 10 cup wins each as players.
Henri Richard with Montreal canadiens - 11 times
Henri Richard won eleven (11) Stanley Cup championships playing as a member of the Montreal Canadiens.
Ken Dryden won 6 Stanley Cups in a career that only spanned 8 years.
Scotty Bowman
The Canadians
Canadiens
ken dryden -6
Montreal Canadians with 24