Trivial Little Baseball Puzzle

I’ve been reading a book about the innovations of baseball so that’s probably why it’s on my mind. And this isn’t important and I don’t expect it to go anywhere, but it did cross my mind, so, why not give it 200 words where they won’t do any harm?

Imagine one half-inning in a baseball game; imagine that there’s no substitutions or injuries or anything requiring the replacement of a batter. Also suppose there are none of those freak events like when a batter hits out of order and the other team doesn’t notice (or pretends not to notice), the sort of things which launch one into the wonderful and strange world of stuff baseball does because they did it that way in 1835 when everyone playing was striving to be a Gentleman.

What’s the maximum number of runs that could be scored while still having at least one player not get a run?

My gut instinct is to say 24, citing the pigeon-hole principle: imagine every player but one gets a run every at-bat; then, there’s eight runs for every out on the unlucky isolated person. Place the unlucky one at the bottom of the order and that’s eight times three before the inning’s out. (Place the unlucky batter first and you get as few was 16.)

Naturally since that answer’s clean, crisp, and obvious, I’m left doubting it. I don’t see a convincing way to get above 24 while preserving one unlucky batter, though I haven’t given this much thought either.