The all time MLB record is 17 consecutive plate appearances reaching base by Frank Ward in 1893. Ward got 8 hits, 8 walks, and hit by a pitch once. The modern day MLB record is held by Hall of Famer Ted Williams who reached base in 16 consecutive plate appearances in 1957. He got six hits, was walked 9 times, and hit by a pitch once.
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Rogers Hornsby had 13 consecutive games with more than one hit in July of 1923.
Joe Dimmagio is the answer for most consecutive games with one or more hits, which is not the question asked.