I posted along these lines elsewhere, but I guess it's worth summarizing: If Ryan LaMarre holds onto his current Twins stats, he will be tied for 14th all-time, in number of PA (8) while maintaining a perfect 1.000 BABIP in a major league season.
The list of leaders is heavily populated by pitchers. Gary Waslewski had an incredible 1970, with 19 PA in 32 games, consisting of 1 hit, 1 walk, 2 sacrifice bunts (which slightly unintuitively don't count as "in play"), and 15 strikeouts.
Among position players, LaMarre is tied for 4th if things don't change. Outfielder Bill Hall for the Orioles holds the record with his 2012 portfolio of 14 PA divided up among 2 hits (one a home run), 5 walks and 7 strikeouts. Those were his last appearances in the major leagues - you would think he must have realized he couldn't improve on perfection and swiftly retired, but actually he played a couple more years in the minors.
The position-player record seems within reach, if Ryan can lace another single or two and make sure to strike out otherwise. Good luck!