How unusual would it be for a major league outfielder to go through an entire game without making a play on any of the 27 (or 24) “outs”?
It’s not unusual at all. I just looked at my scorebook for this season. In the first nine games it happened to seven outfielders. A couple of times it happened to both teams.
What I haven’t found so far is an instance of there being no outs by a center fielder.
Thanks, was it mostly the right fielder?
In five cases it was the left fielder, and in two it was the right fielder.
I think this is too small a data set to make any meaningful extrapolation.
In the interest of science I looked up Blue Jay box scores this year starting right from Opening Day. On Opening Day, neither the right fielder (Jose Bautista) OR the left fielder (Michael Saunders) is credited with a putout or an assist. Bautista did make an error, though.
In Game 3, neither Bautista nor the center fielder, Kevin Pillar, made any outs.
I scanned through about 40 games and lost count eventually, but it’s very common.
I wonder if there is a way to determine if a fielder didn’t make any plays at all (not just put outs or assists). It would probably require a game with a lot of strikeouts, and maybe a no-hitter.
Are the fielders that touched the ball on a non-out play recorded anywhere?
Looks like in Kerry Wood’s 20 K game the left fielder never handled the ball.
This is noted in the detailed scoresheets but it’s not an official statistic, so it takes some digging.
Wait, so this is why my Little League coach played me in left field! 
Think that’s strange? How about an entire game where the first baseman doesn’t record an out!
Something like that could never happen twice.
In fact, the odds against it are something like the odds of lightning striking 19 times.
Ortiz had an assist in the game mentioned in your first link, and Cash (second link) played just eight innings. Amateurs! 
The game I remember is Gene Tenace with Oakland in 1974, when he had no putouts and no assists in a game against Detroit that went the full nine innings. (According to your third link, there have been a handful of others that fill the same bill.)
Anyway, I looked up the Tenace game on Retrosheet, Retrosheet Boxscore: Oakland Athletics 5, Detroit Tigers 3, and I suppose it’s conceivable that he never touched the ball even once while it was live and in play. You can’t be sure from an account like this, of course.
–There could have been an unsuccessful toss over to first base to try to pick off a runner. Noted basestealer Ron LeFlore was one of a few Tigers who reached first during the game. I’d actually be more surprised if there wasn’t such an attempt than if there was, but retrosheet only logs it if the runner was tagged out or there was an error.
–Singles to the outfield–the Tigers had several in this game–are sometimes followed by a throw over to first, if the batter made a wider turn than usual. No indication that this happened, but no reason to think it didn’t.
–There were a couple of force plays at second, such as Bill Freehan forcing a runner on a grounder to short. Easy to imagine the second baseman trying the relay throw to Tenace, especially given Freehan’s lack of speed, only the throw arrived late.
–First basemen are sometimes cutoff men on throws from the right fielder or center fielder. There were several points where that could’ve happened.
So…probably not. But it is fun to think about.
I thought all the stiffs got put in right field. I couldn’t catch a high fly ball to save my life.
I think the ultimate record might be that of Texas Rangers shortstop Toby Harrah, who once played an entire double-header at shortstop with no assists, putouts, or errors.
Random cite: Toby Harrah - Wikipedia
That is the most astounding baseball stat I think I’ve ever seen.