Of the non-original six teams only Edmonton has more cups than the Pens, with 5 cups.
The Rangers, an original six team, is tied with 4 cups. as are the Islanders.
Since 1967, the teams who have won the most cups are:
Montreal - 10 (although that should have an asterisk as well as they basically owned the Quebec pipeline for the decade after expansion, when most of these took place. None in a quarter century).
Edmonton -5
Pittsburgh - 4
NYI - 4
Detroit - 4
Boston -3
Chicago - 3
Since the original six in the 20's, the most cups are:
Montreal Canadiens: 23
(The Canadiens also have one pre-NHL win, listed below)
1924, 1930, 1931, 1944, 1946, 1953, 1956, 1957, 1958, 1959, 1960, 1965, 1966, 1968, 1969, 1971, 1973, 1976, 1977, 1978, 1979, 1986, 1993
Toronto Maple Leafs: 13
(Includes wins under previous franchise names: Toronto Arenas and Toronto St. Pats)
1918, 1922, 1932, 1942, 1945, 1947, 1948, 1949, 1951, 1962, 1963, 1964, 1967
Detroit Red Wings: 11
1936, 1937, 1943, 1950, 1952, 1954, 1955, 1997, 1998, 2002, 2008
Boston Bruins: 6
1929, 1939, 1941, 1970, 1972, 2011
Chicago Blackhawks: 6
1934, 1938, 1961, 2010, 2012, 2013
Edmonton Oilers: 5
1984, 1985, 1987, 1988, 1990
Pittsburgh Penguins: 4
1991, 1992, 2009, 2016
New York Islanders:4
1980-83
New York Rangers: 4
1928, 1933, 1940, 1994
http://proicehockey.about.com/od/stanleycupbunker/a/stanley-cup-wins-team.htm
Let's do this tonight. 
