Is there anything “special” about a second application of shampoo, as opposed to one application, or as opposed to just leaving it on your hair longer before rinsing? How about between a second and third application?
Note: this is in GQ versus IMHO because I’m looking for a scientific/factual answer to the question of how/why/if the second application of shampoo operates, rather than a discussion of people’s experiences with shampoo application.
It probably depends on your hair’s oiliness. For me, absolutely. The first wash doesn’t really do anything, barely lathers. But it is stripping away the oil. Then the second one foams up and feels squeaky clean.
Dennis
Note that those oils are actually good for your hair’s health.
Altho the “No Poo” movement is going too far, once a week or anytime after you get your hair dirty is fine.
For a week or two your hair will feel odd. Stick with it.
ALWAYS repeat!
Indeed they are. When I was a much younger man, I spent a month camping in the Wind River Mountains in Wyoming, as part of a National Outdoor Leadership School course. Since we were deep in the Bridger Wilderness backcountry, that lacked even defined trails, bathing and showering went into abeyance for that time. Everyone complained about their own greasy hair and reeking armpits, but after the first shower when we returned to civilization, all the women on the course commented on how full and shiny their hair was, and how easy to manage.
For my part, that month was the only time that my beard ever felt as soft as the hair on my head; generally, it was wiry and bristly.
At a minimum, it can double sales of shampoo - if you get stuck in an infinite loop, even more sales 
The oils may be preventing anti-dandruff treatment getting in to do its job.
Otherwise its just the amount of oil you want to strip out… if you want a dry look ?
You might get away with a one application daily shampoo if you do shower daily.
But it must be the cause of bad hair days.
The infinite loop always ends with a hot water underflow exception.
I always figured it was there mostly to increase sales, as few people actually need to wash their hair twice in the same shower. With other toiletries (shaving cream and toothpaste, for example) they boost sales by showing people using far more that they actually need.
It improves the softness of your leather wallet, as there is less cash in there to suck up all the leather’s oil and dry it out.
Yep, this is about as close as we’ll get to an objective reason for the instruction.
I try to use the absolute minimum. If one pass doesn’t feel right, the second always does.
Oh Man, I would look and feel horrible if I skipped more than a day. Anyway, I’m 71 years old with a full head of hair, not even a receding hairline, and not even all gray. Seems to work OK. I have been using Head and Shoulders all this time (actually the Walmart generic version for the past 20 years).
Dennis
Yeah I read your note. Didn’t think it was going to work.
Vinegar works better on dandruff than any of the specialty shampoos.
Leaving it in doesn’t make it lather any better. And since it lathers better the second time, there must be some difference in the two different applications
Now whether that difference means it’s actually cleaning more, I don’t know. It’s possible that cleaner hair lathers better.
I do have my personal experience, where my hair still winds up oily after the first time, but I also don’t wash my hair all that often. Plus it is long. I seem to remember it being less of an issue when my hair was short and I washed it every day.
Next time, more stealth 
I still say the repeat instruction dates from the time when people didn’t wash their hair every day. I was alive then, and the schedule imposed on me for hair washing was Monday, Wednesday, and Saturday nights. The second lathering was quite necessary. Later, when I started washing it every morning by choice, the repeat became unnecessary.
As a poor college student I discovered Dawn dishwashing detergent makes for a passable shampoo, but only once or twice a week. It’s a little on the harsh side. Ok a lot on the harsh side.
If I figure out the amount of shampoo (S) it takes to get my hair clean in one go, then instead use a lather/rinse/repeat cycle, I can get it clean with less shampoo. I’ll use 1/4 S the first time, which doesn’t seem like it’s doing anything, then I’ll rinse it out and repeat with 1/2 S. This lathers up perfectly, and gets my hair clean. So I’d say you can save money, as long as you don’t hold the ridiculous assumption that you should use the ‘full’ amount for a partial wash.
I don’t normally do this double wash, because time is more of a consideration than shampoo cost. But when my hair is really dirty I know that virtually no amount of shampoo is going to be sufficient in a single wash, so I do the prewash.