ADDED 4/24/17: I’m leaving this post up, but Brandon no longer holds the record…after failing to score a run through his first 115 career plate appearances, he finally touched home plate on April 23, 2017.
On April 19, 2015, Dodgers pitcher Brandon McCarthy led off the bottom of the third inning and struck out. That gave him sole possession of an all-time major league record. It was the 89th plate appearance of McCarthy’s career, and he has never scored a run. That broke a tie with 1980s Montreal Expo Razor Shines, who finished his career runless in 88 trips to the plate. I have a bit more about Razor at the end of this post. McCarthy had a sacrifice bunt in his other plate appearance in the game to increase his record of runless PAs to 90. (ADDED 3/24/16: McCarthy made only one more start in 2015 before his season ended due to injury. He enters the 2016 season with 93 career plate appearances — and no runs scored.)
McCarthy didn’t even reach base in his 10 trips to the plate in interleague games during his years in the American League from 2005-12. He hasn’t done much with the bat in his hand since coming to the National League, with just five hits (all singles) and four walks (he also reached on an error once). He’s never reached third base (although to be honest I haven’t checked to see if he ever reached base on a fielder’s choice and advanced to third after that).
Here is the list, updated through 4/19/2015, of all players in major league history who have had at least 50 career plate appearances without scoring a run. Note almost all of these players were primarily pitchers, the exceptions being Shines and two players from the deadball era, catcher Joe Jenkins and outfielder Frank Hemphill.