What is the record for most consecutive balls (not intentional) thrown by a pitcher in the MLB?
Most likely, this record is unknowable, as Major League Baseball has only kept pitch-by-pitch records since 1988.
Given the much lower quality of play in the early 1900s, games such as the Washington Senators' loss to the Chicago White Sox in 1909, where the box score records 11 walks by Dolly Gray, eight of which were in a single inning, would be a good candidate - eight straight walks seem to likely have a pretty long strikeless streak in them (especially as players quickly learn to stop swinging at pitches).
However, the modern-day candidate for this record is from a game between Kansas City and the Chicago White Sox in 2002. In the 8th inning, Miguel Ascencio came in to pitch already down 9-0, in his major league debut, and walked (to lead off the inning!) Kenny Lofton, Ray Durham, Frank Thomas, and Magglio Ordoñez, all on 4 pitches, for 16 consecutive balls. He was then removed, and Paul Konerko was walked on a 4-1 count by the pitcher to follow him, Cory Bailey; that went 3-0, then a called strike, then another ball. (One note: MLB doesn't really differentiate intentional balls from accidental ones, with the sole exception of intentional walks; none of these were recorded as an intentional walk.)
So depending on whether you're looking at pitches thrown by a single pitcher or not, 16 or 19 is a pretty good candidate for it. But it's very possible this was not the record, and instead Dolly Gray or someone else from the early 1900s holds the actual record - but that will never be known.