Four consecutive World Series championships: Yankees 1936,1937,1938,1939.

Before 1936 no team had won more than two consecutive World Series championships or more than three consecutive pennants:

Three consecutive pennants 1903-2020. Four. FIVE! Tuesday, January 5, 2021

At least three consecutive pennants (World Series win BOLD) has been done 16 times, 8 by the Yankees, the only team with a streak greater than 3: 5 twice, 4 twice.

At least three consecutive World Series championships:
5: Yankees: 1949-1953
4: Yankees: 1936-1939
3: Yankees: 1998-2000
3: Athletics: 1972-1974

		1	2	3	4	5	Pen	WS	WS con			Managers
Chicago	Cubs	1906	1907	1908			3	2	2	Cubs	1	Frank Chance
Detroit	Tigers	1907	1908	1909			3	0	0	Tigers	2	Hughie Jennings
New York	Giants	1911	1912	1913			3	0	0	Giants	3	John McGraw
Philadelphia	Athletics	1913	1914	1915			3	0	0	Athletics	4	Connie Mack
New York	Giants	1921	1922	1923			3	2	2	Giants	5	John McGraw
New York	Yankees	1921	1922	1923			3	1	1	Yankees	6	Miller Huggins
New York	Yankees	1926	1927	1928			3	2	2	Yankees	7	Miller Huggins
Philadelphia	Athletics	1929	1930	1931			3	2	2	Athletics	8	Connie Mack

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In 1936 Joe DiMaggio joined the Yankees and they immediately won the pennant and World Series four consecutive times. Joe McCarthy was the Yankee manager 1931-1946. Data from https://www.baseball-reference.com/

1939 World Series	4-0	New York Yankees (106-45, AL) vs. Cincinnati Reds (97-57, NL)
1938 World Series	4-0	New York Yankees (99-53, AL) vs. Chicago Cubs (89-63, NL)
1937 World Series	4-1	New York Yankees (102-52, AL) vs. New York Giants (95-57, NL)
1936 World Series	4-2	New York Yankees (102-51, AL) vs. New York Giants (92-62, NL)

Year (Bold=Pennant)	C	1B	2B	3B	SS	LF	CF	RF	DH
1940  (88-66, 29.2 oWAR)	Dickey (1.3)	Dahlgren (0.1)	Gordon (6.1)	Rolfe (1.5)	Crosetti (-0.2)	Selkirk (3.4)	DiMaggio (7.3)	Keller (5.6)
1939  (106-45, 42.9 oWAR)	Dickey (5.5)	Dahlgren (-0.4)	Gordon (6.3)	Rolfe (6.6)	Crosetti (2.1)	Selkirk (5.8)	DiMaggio (8.4)	Keller (4.9)
1938  (99-53, 30.4 oWAR)	Dickey (5.2)	Gehrig (4.7)	Gordon (3.5)	Rolfe (4.4)	Crosetti (3.6)	Selkirk (1.1)	DiMaggio (5.8)	Henrich (3.8)
1937  (102-52, 30.6 oWAR)	Dickey (6.7)	Gehrig (8.3)	Lazzeri (1.6)	Rolfe (2.8)	Crosetti (1.0)	Powell (-0.4)	DiMaggio (8.3)	Selkirk (3.2)
1936  (102-51, 37.6 oWAR)	Dickey (5.9)	Gehrig (9.7)	Lazzeri (2.3)	Rolfe (4.9)	Crosetti (3.9)	DiMaggio (4.6)	Powell (1.1)	Selkirk (4.2)
1935  (89-60, 31.7 oWAR)	Dickey (3.0)	Gehrig (8.7)	Lazzeri (2.6)	Rolfe (4.1)	Crosetti (2.6)	Hill (1.7)	Chapman (3.6)	Selkirk (4.1)

Year	Starting Pitchers					Closer	Bullpen
1939 (106-45)	Gomez	Donald	Hadley	Pearson	Ruffing	Murphy	Chandler	Sundra	Russo
1938 (99-53)	Gomez	Chandler	Hadley	Pearson	Ruffing	Murphy	Andrews	Sundra	Hadley
1937 (102-52)	Gomez	Wicker	Hadley	Pearson	Ruffing	Murphy	Andrews	Makosky	Malone
1936 (102-51)	Gomez	Broaca	Hadley	Pearson	Ruffing	Malone	Murphy	Kleinhans	Brown

Joe DiMaggio
1939 Play Ball cards, published by Bowman Gum
Public domain, via Wikimedia Commons