New York City is the only market that has ever had three teams in the same major league in any sport at one time.
From 1903-57, it was in Major League Baseball. Manhattan had the Giants, Brooklyn had the Dodgers, and the Bronx had the Yankees. Only the Yankees are still in NYC, the NL currently has the Mets in Queens, but someday in the future I could see Brooklyn getting its own MLB team again (but other markets would have to get teams first especially if we're gonna realign MLB into 8 divisions of four teams each that all make geographic sense, and if Brooklyn does get another MLB team it would have to be in the NL for tradition's sake).
NYC currently has three NHL teams. The Rangers have always been the top team in the five boroughs, and the northern suburbs in NY and CT. The Islanders main fan base has been suburban Long Island. The Devils primarily have their fans on the New Jersey side of the metro area.
The three NYC area teams have always shared a division and thus played each other multiple teams a year (in contrast, the Yankees could only play the Dodgers or Giants in the World Series for many years, and regular season games between the Yankees and Mets did not arrive until 1997).
Thus, you could say Devils-Islanders-Rangers is a huge three-way rivalry.