The only team in Major League Baseball history to win 5 consecutive World Series titles is the 1949-1953 New York Yankees. Led by manager Casey Stengel, the team had a different look from 1949 until the last of the consecutive titles. This happened right during one of the biggest transitions in the history of the franchise; Joe DiMaggio going out and Mickey Mantle coming in. Remember, when Babe Ruth left to join the Boston Braves after the 1934 season, the Yankees were not coming off a World Series appearance (the St Louis Cardinals beat the Detroit Tigers). It is even more amazing as it seems like the team didn't miss a beat as they continued to win.
Obviously, there was one player that stood out and should be considered the heart and sole of that team and that was their catcher, Yogi Berra. Without him, the Yankees do not beat the Brooklyn Dodgers, Philadelphia Phillies, New York Giants and the Dodgers two more times.
The Yankees had three pitchers who were part of all five WS Championships and one of them was not Whitey Ford. Ford made his debut in 1950 and missed 1951 and 1952 due to military service. Vic Raschi won 21 games in 1949, 1950 and 1951 and started at least a game in each one of the World Series. He won a game in each of the series for the exception of the 1953 series against the Dodgers. An older Allie Reynolds did win a game in each of the five World Series and along with Raschi were the backbone of the pitching staff. The left hander, Eddie Lopat was the other pitcher to be part of all five teams, also starting games in all five World Series.
Jerry Coleman was a player that was as deserving as any to be part of all five of those teams. While he did play games for the Yankees in each one of those seasons, he served in the military in both 1952 and 1953 and did not play in each of those World Series. He came back in 1954 for the Yankees and would play in three more World Series (1955-1957). 1B Joe Collins also appeared on all five of those teams, but just had a cup of coffee on the 1949 team, just 10 AB. He made his World Series debut a year later, getting into just one game without an at bat. He would be an important part of the next three, as well as in the 1955 and 1956 series, then getting into the 1957 fall classic as a pinch hitter in 6 games.
The next player to be part of all five teams already had a Hall of Fame career before he ever joined the Yankees. He had also never appeared in the postseason before he arrived at Yankee Stadium. Johnny Mize was purchased late in the 1949 season by the Yankees from the Giants and had 23 regular season ABs for them. He went 2-2 in the World Series, driving in a pair before remaining with the Yankees for the final 4 seasons of his career. A man who put up some of the most consistent numbers for a first baseman of his time got to finish off his career by winning 5 consecutive World Series Championships.
To finish off the answer to this trivia question, the other two players to be on all five teams were catchers Ralph Houk, who got a total of 36 at bats in 5 regular seasons with the Yankees and just 1 AB in the World Series and Charlie Silvera, who would get a couple more at bats with the team (between 25-130 over the course of the five seasons) as well as 2 ABs in the 1949 WS. This makes it a total of 12 players to appear on all five teams.
Among those to play for the 5 straight winners, 4 players also appeared on the 1947 World Series Champion team led by manager Bucky Harris. Berra, Rizzuto, Raschi and Reynolds all had ties to that team. Bauer would add WS titles with the Yankees in 1956 and 1958, while Coleman, Rizzuto and Collins would win with the Yankees in 1956. Of course, Berra would win a total of 10, on the years of 1947, 1949-1953, 1956, 1958, 1961 and 1962.