Let's take a look at the tape (courtesy Baseball Almanac (dates the current team and previous names of that franchise played are given)...
Chicago Cubs 1902-present
Chicago Orphans 1898-1901Boston Red Sox 1908-present
Chicago Colts 1890-1897
Chicago White Stockings 1876-1889 (note; the National League formed in 1876)
Boston Americans 1901-1907 (note; the American League formed in 1901)New York Yankees 1913-present
New York Highlanders 1903-1912Now for some other teams that have been around a while in the same city...
Baltimore Orioles 1901-1902 (sorry Yankee fans, but I have to call them as I see them)
Chicago White Sox 1904-present
Chicago White Stockings 1901-1903Cincinnati Reds 1959-present
Cincinnati Redlegs 1953-1958Cleveland Indians 1915-present
Cincinnati Reds 1890-1952
Cincinnati Red Stockings (American Association) 1882-1889
Note; A team alternatetly called the Cincinnati Red Stockings and the Cincinnati Reds were in the National League between 1876 and 1880 before being kicked out because they refused to stop selling beer and refused to stop playing on Sunday. Also, the Reds are generally accepted to have been formed as a franchise in 1871.
Cleveland Naps 1905-1914Detroit Tigers 1901-present
Cleveland Blues 1901-1904
Philadelphia Phillies 1890-present
Philadelphia Quakers 1883-1889Pittsburgh Pirates 1891-present
Pittsburgh Alleghenys 1887-1890St. Louis Cardinals 1900-present
Pittsburgh Alleghenys (American Association) 1882-1886
St. Louis Perfectos 1899
St. Louis Browns 1892-1898
St. Louis Browns (American Association) 1883-1891
St. Louis Brown Stockings (American Association) 1882