These villages on Long Island:
Head of the Harbor
Village of the Branch
North Great River
North Bay Shore
Shelter Island Heights
Cold Spring Harbor
Great Neck Estates
Garden City Park
Malverne Park Oaks
Fine, we’ll stick to the theme.
Honorable mention to Niagara-on-the-Lake. It’s actually in Canada but right on the border.
Near me there’s a Sixteen Mile Stand and a Mt. Healthy Heights, but they’re just census designated places, not incorporated municipalities so boo. Still, I wouldn’t say Mt. Healthy Heights necessarily follows the upper/lower or north/south exclusion per se. There is a city called Mt. Healthy nearby, but their names don’t play on one another. Similarly there’s a legit North College Hill (an independent municipality as distinct from the Cincinnati neighborhood College Hill), so there isn’t a South College Hill or any other municipality with College Hill in its name.
Here’s one to mull over. In Ohio, incorporated municipalities are either villages or cities. A village automatically becomes a city when the population reaches 5,000 and vice versa. Well, the residents of Indian Hill wanted to pretend they’re still a village even though the population grew above 5,000 by 1970. So they renamed the city “The Village of Indian Hill” which technically means it’s “The City of The Village of Indian Hill”, showing up under either T or V in alphabetical listings.
I was beaten to Washington Court House, oh well.
I considered adding North College Hill and Indian Hill but I figured they didn’t meet the OP’s limitations.
Indian Hill would have to count considering its actual name is five words.
As a technicality only. Everyone calls it “Indian Hill.” That’s what its name is meant to be. But also, I would consider “the Village of” to constitute a descriptive element ruled out by the OP.
According to Wikipedia’s list of US cities by population, the largest (by population) three-word placename in the USA would be the country’s 79th largest city, North Las Vegas, Nevada, at about 250,000. If we exclude that because of “North” being similar to the excluded “Upper” (and because it’s generally thought to be part of metropolitan Las Vegas, I suppose), then the next in line is Port St. Lucie, Florida (population 205,000, 105th largest city in the USA).
I wouldn’t count cases where (as in Los Angeles) the polysyllabic name is only historical and neither commonly used nor official under current law, but it’s surely possible to disagree on this point.
Also, Carmel-by-the-Sea and Half Moon Bay, both in California. And Croton-on-Hudson, New York.
South Lake Tahoe, CA.
Ciry of Industry, CA.
Rancho Palos Verdes, CA.
Santa Fe Springs, CA.
But isn’t technically correct the best kind of correct? 
Technically any village can be called “The Village of…” or any city can be called “The City of…”, but the fact that they added “The Village of” to the actual name, despite it being a city, means it’s not a descriptive element, because it’s wrong.
Truth or Consequences, NM
Is its complete and official name “The City of The Village of Indian Hill”? In that case, I’d say that the “The Village of…” part is not a technicality but part of the actual name. If the complete name is, however, “The Village of Indian Hill”, then I’d say that the “The Village of…” part is a technicality, analogous to the “City of…” part in the official names of many other cities, even if it is wrong in substance. A similar case, in Canada, is Dawson City, Yukon, which is officially called the City of Dawson although it has been downgraded from a city to a town.
It’s still a descriptive element. And “only technically correct” to me is of a lesser class than “correct.” I consider bureaucratic shenanigans to be less definitive than actual usage.
For example, back when the official name of Washington Court House was “the City of Washington,” I considered the “official” name less correct than the commonly used name.
Being the OP, i would like to exclude any geographic terms in the city/town names - thus rejecting most of those Long Island names.
According to Wikipedia yes, but you know how that goes. I don’t think that in Ohio “The City of” or “The Village of” are required as part of the complete and official name, they’re just prepended automatically in much the same way as Mr. or Mrs. It’s like Indian Hill got married, but still wanted to be called Miss instead of Mrs., so they added Miss to their given name, leading to Mrs. “Miss Indian Hill”.
Palos Verdes Estates, CA
Florida has them by the dozen, even if you delete ones with “North” or “Village.”
You can’t do better than Sea Ranch Lakes, which is an enclave inside Lauderdale-by-the-Sea.
BTW, the OP makes a mistake so common that even I’ve done it. There is no place named New York City. It’s just New York. People automatically add “City” to distinguish it from New York State.
Are apostrophe’d and contracted names acceptable? Does Coeur d’Alene in Idaho count?
And Town and Country, MO abuts Crystal Lake Park and completely surrounds Country Life Acres.
Just a few miles north lies Portage des Sioux.