It is illegal to climb Gangkhar Puensum in Bhutan, so no-one has ever reached the top. It is the highest unclimbed mountain in the world.
The mountain Gangkhar Puensum (“White Peak of the Three Spiritual Brothers”) in the Himalayas is 7,570 metres high. There are only thirty-nine higher peaks in the world, all of them in the Himalayas or the nearby Karakorum. That means Gangkhar Puensum is higher than every mountain in the Americas, Africa, and Europe. And no-one has ever climbed to the top.
Gangkhar Puensum is not a notorious technical challenge to climb. It is remote, but not nearly as difficult to get to as Mount Siple (probably the highest unclimbed mountain in Antarctica). No, the challenge of Gangkhar Puensum is its nationality. The peak is located mostly or entirely within Bhutan. In Bhutan, it is illegal to climb any mountain higher than 6000 metres.
For the Bhutanese, the mountains are sacred and standing on top of them is profoundly disrespectful. (The same is true of Uluru in Australia, although climbing that one was only banned in 2019!) The highest Bhutanese peaks were open to climbers before 1994, and several expeditions tried to climb Gangkhar Puensum, but none succeeded in time.
Japanese climbers were going to attempt the summit in 1999, and sidestepped Bhutan by appealing instead to China. The exact position of the border between Bhutan and Tibet is disputed, and the Chinese hold that Gangkhar Puensum is halfway inside their territory. They were all set to go, but at the last moment Bhutan protested.
Presumably to avoid a diplomatic incident, the team shifted their target to Liankang Kangri. It’s basically the same mountain: Liankang Kangri is only 35 metres lower than Gangkhar Puensum, and a 2km-long ridge connects the two. But the lower peak supposedly sits entirely within Tibet, so Bhutan could not protest. And Gangkhar Puensum retains its title as the highest unclimbed mountain in the world.