My former business partner claims that his family owns a mountain in Pennsylvania that is used for mineral exploitation, if recall well. Of course, the question is whether this elevation is literally high enough to be called a mountain.
Halfway there:
This couple in North Carolina bought half a mountain (i.e., the western side), and built very nice cabins for vacationers. Not sure how high the mountain was. The cabins were super. Name on request.
I was just talking about Corbin Park this weekend - I think you can hunt wild boar there. But yes, that’s not only private but very controlled access due to the hunting.
This is at 45° 16’ 41.86" N 111° 27’ 1.84" W? The western face is actually national forest, about 1/8 of it towards the top and up to 1/3 down lower. There’s also a section owned by “Big Sky Resort Inc” as a separate owner, not sure what the legal distinction of separate incorporpation is here.
Surgarloaf mountain in Maryland is owned by the Stronghold Inc. It’s open to the public and it is quite popular for hiking.
There are no mountains around here, but I know the woman who owns the Kleine Bestener See, about 30 km (20 mi.) south of Berlin. It is a smallish lake, perhaps 30 acres in total, there are about 11 similar lakes in the district of Bestensee alone, and hundreds if not thousands in Brandenburg and Mecklenburg-Vorpommern (never mind in Finland or Sweden, but those may be state owned and free to access for the public).
The woman in question is not particularly affluent, the lake was used for low level aquaculture (pike-perch, carp and the like).
Spain has some really big land owners, particularly in the south half. I guess some of them own not a mountain, but a whole mountain range. I am not going to search for the details, it would be bad for my blood pressure. Just google “Familia Mora Figueroa” or “Familia Domencq” and “EU subsidies for agriculture” or “Miguel Arias Cañete, EU Commission”.
In Germany the family Thurn und Taxis owns the half of Bavaria that is not owned by the Catholic Church (I’m not exagerating by much).
The western face of Lone Mountain is “sidecountry”, and is USFS Wilderness, so, yes, Big Sky does not own the whole mountain, just 2/3 of it.
There are several 14’ers (mountains over 14,000 ft elevation) in Colorado that are privately owned.