This is an exercise I've done a few times, generally within the comment section. As it currently stands, I believe that almost every major leaguer is within six degrees of the Bambino (with a degree defined as another person with whom they appeared in the same game.) As a rule, Bartolo Colon and Jim Kaat have a lot of connections, given the length of their careers. I'll try to avoid both of them.
Connection to Ruth
First degree: Asdrubal Cabrera. Cabrera's career dates back to 2007, when he was part of a surprising Indians team that fell just short of the World Series, giving up a 3-1 series lead over the Red Sox. In 2008 he was a regular middle infielder, starting at both shortstop and second base. And it was in May of 2008 that he appeared in the same game as the next link in the chain.
Second degree: Frank Thomas. The Big Hurt was in the last season of a Hall of Fame career that saw him hit over 500 home runs while batting over .300 (not to mention his career OBP of .419.) His debut came on August 2, 1990, in a game which featured three more Hall of Famers and two more members of the 500 home run club that haven't been elected. He appeared in the White Sox' lineup between Carlton Fisk and Sammy Sosa, while the Milwaukee Brewers lineup featured Robin Yount, Paul Molitor, and Gary Sheffield.
Third degree: Carlton Fisk. Fisk made his debut in 1969, but wasn't a regular player until 1972. He may have been the first catcher nicknamed "Pudge" and certainly redefined the catcher position. He also continued to catch regularly into his forties, and was the starting All Star catcher in 1991 at the age of 43. 1991 was also just the third season of his career with below-average OPS.
Fourth degree: Tony Conigliaro. Conigliaro's career was cut short by being hit in the eye by a pitch. He missed the entirety of the 1968 season as a result, was never the same player after returning. He remained an above-average bat in 1969 and 1970, but after a poor 1971, was out of the major leagues from 1972-1974, before coming back to Boston for 69 plate appearances in 1975. He played with Fisk in both 1969 and 1975.
Fifth degree: Satchel Paige. On September 25, 1965, Charlie Finley had one of his better ideas to boost attendance, bringing the legend back to start against the Red Sox. Reportedly, when Finley asked if he could pitch three innings, Paige responded "how many times a day?" (Clearly, today's overly coddled and still constantly injured pitchers could stand to learn a thing or two.) Paige worked his three innings (allowing no runs and just one hit, to Carl Yastrzemski, who also played with Pudge Fisk) before he was removed from the game.
Sixth degree: Babe Ruth. Paige supposedly struck out Babe Ruth four times, when both were playing in the California Winter League. Unfortunately, we were robbed of the two facing each other at their respective peaks, thanks to segregation.
Do I detect a note of protest that Paige and Ruth never played in the same major league game? Ok, but that's no fun. For starters, they should have played in the same game (unless Paige signed with a National League team) and allowing Paige to count makes the exercise more fun. But, for those who don't like fun...
Tyler Gilbert's debut came on August 3, as he picked up a hold and Tyler Clippard got the save. Clippard's debut came on May 20, 2007, as he pitched six innings against the Mets and got the win. Mariano Rivera struck out Julio Franco to end that game. As a rookie in 1982, Franco faced Jim Kaat, who had faced Ted Williams as a rookie in 1959. Williams faced Red Ruffing in his debut, who had been a teammate and opponent of Ruth.
Connection to Bobo Holloman
The only previous modern-era pitcher to throw a no-hitter in his first start, Holloman was a teammate of Satchel Paige, albeit briefly. He was also a teammate of Don Larsen.
In his no hitter, he faced the Athletics and their third baseman, Loren Babe. Babe had started the year with the Yankees, who were starting Mickey Mantle in center field. In 1968, Mantle was playing first base, and the other corner of the Yankees infield was occupied by one Bobby Cox, who was Franco's last manager, and also the first manager of several players active today, including Jason Heyward and Craig Kimbrel.