Rickey Henderson was on a previous iteration of this quiz, but he was the least guessed so I removed him. He also changed teams about 10 different times.
Johnny Damon, Pedro Martinez, Wade Boggs, and Roger Clemens all left as free agents. The Red Sox offered Clemens a ton of money, but he turned out it down. They had no chance but to trade Manny Ramirez, who basically threw a tantrum and refused to play anymore...because he is an awful person. The Babe Ruth trade...well...that was bad. The only remarkable thing about the other five exits is that both Boggs and Damon went to the arch-rival Yankees.
Giambi really doesn't belong here... everyone else is in the HOF, or would be if not for steroids, or will be in the case of the active players, or has an iconic moment of baseball history associated with them. Giambi is none of those. I'd love to instead see Hank Aaron, who left the Braves for the Brewers at the end of his career.
Giambi was the 2000 AL MVP and had a better season in 2001 (by WAR), only to miss another award due to Ichiro stealing the show. He then joined the team that beat Oakland two years in a row in hard-fought playoff series, badmouthing the team and fans on his way out. It was a huge deal in baseball at the time.
Gibson, meanwhile, was successful in Detroit prior to signing with the Dodgers, winning a World Series and having a reputation as a slugger with speed. He then immediately won the NL MVP in LA and hit the most-replayed homer in World Series history on the way to winning the title. Another significant move at the time.
Tom Seaver was a pretty high-profile player to change teams. Threw the only no-hitter of his career with Cincinnati, and should have won the Cy Young in '81 with them.
But I'll take your suggestion and add Wade Boggs.
Giambi was the 2000 AL MVP and had a better season in 2001 (by WAR), only to miss another award due to Ichiro stealing the show. He then joined the team that beat Oakland two years in a row in hard-fought playoff series, badmouthing the team and fans on his way out. It was a huge deal in baseball at the time.
Gibson, meanwhile, was successful in Detroit prior to signing with the Dodgers, winning a World Series and having a reputation as a slugger with speed. He then immediately won the NL MVP in LA and hit the most-replayed homer in World Series history on the way to winning the title. Another significant move at the time.