Back in the middle of the previous century, a brave man called Edmund Hillary from New Zealand is said to have reached the highest available summit on he Earth, accompanied by his Nepali fellow, Tenzing Norgay. Mountaineering equipment and technique has undergone an unbelievable development since then. Even a 13 years old child, a man who had his feet amputated, or a blind mountaineer could reach Mount Everest in the latest decades. We could think that there are no real challenges remained for mountain climbers.
This is of course not true. There are numerous peaks found on our planet that were not climbed by humans in our history. There are so many such summits, that the number of them is simply unknown. There could several reasons why a mountain remains unconquered. Let's look at some of the most famous examples.
The highest peak that has not been climbed is the so called Gangkhar Puensum (sometimes also spelled as Gankar Punzum or Gangkar Punsum). This peak is also the highest mountain in Bhutan. Mountains in Bhutan were not open to mountaineers (or anyone else) before 1983. There were a couple of unsuccessful expedition in the next years, and finally in 1994, access to all mountains higher than 6000 has been closed in order to protect the sacred homes of the deities and spirits where no humans shall pass. So the peak remained unclimbed, and is not expected to be conquered in the near future.
The highest unclimbed mountain where climbing is allowed is probably Pakistan's Muchu Chhish of the Karakoram range, with a height of 7452 m, and a topographic prominence of 263 m. Despite a couple of attempts to reach the summit, noone was ever able to do it actually.
Labuche Kang III East in Tibet, China, is also a notable peak that has zero successful climbing attempts. It has an elevation of 7250 m and a prominence of 570m.
Sauyr Zhotasy (or Mouz Tau) might not be the highest summit top of the list, as it only reaches roughly 3840 meters (12,600 ft) above sea level, but it has a prominence of 3252 m (or 10,669 ft), making it the most prominent unconquered mountain of the planet. At least there are no records showing that anyone has climbed it. We cannot be sure with a mountain of such a relatively low height. The peak is located on Chinese territory, close the the border with Kazakhstan.
Saser Kangri II East is the odd one out on this list. Saser Kangri, located on the northern corner of India, reaches 7518 m with a prominence of 1450 m (24,665 ft / 4,757 ft). The reason why it is the odd one out is that it was an unclimbed peak. It has been climbed in 2011 by Mark Richey, Steve Swenson and Freddie Wilkinson, deleting it from the list of unclimbed mountains. Before that date it was said to be the second highest unconquered peak.