Mount Kailash (literal translation of the Devangiri Kailāśā Parvata) is the greatest mountain on Earth never to have been climbed. The reason is simple: it is also the most sacred mountain, and one of the most sacred sites, in the world, having central importance in no fewer than four religions: Hinduism, Buddhism, Jainism, and the ancient Tibetan religious tradition of Bön. Climbing it - in fact, setting foot on the peak at all - would be considered by members of any of these religions to be sacrilege, and since there are always pilgrims at the foot of the mountain, any attempt could not go unnoticed.
Mount Kailash is part of the Gangdise mountain range, a part of the Tibetan Himalayas that is the source of three of the longest rivers in Asia - the Indus, the Brahmaputra, and the Sutlej. It is also the source of the Karnali, one of the rivers feeding the Ganges. At 6,638 m it isn't one of the tallest mountains in the world, but it is nevertheless an extremely striking peak visually, with snow-covered high slopes rising starkly for 1.5km from the surrounding plains in an unusually perfect 4-faced dome. It can be travelled to from various different cities in China, India or Tibet, and once there, a pilgrim or tourist will be staying in the settlement of Darchen, a mostly unregulated camp in which you need a Chinese permit (or a decent bribe) to stay.
In Hindu myth, Shiva resides at the summit of Mount Kailash with his wife Parvati, in continuous meditation. It has been variously described as the center of the world, Shiva's home, Shiva's lingam, or the pillar of the world, and has been described in over-the-top detail in various Hindu texts as having slopes of jewels, gold and lapis lazuli, and as being paradise on Earth.
In Buddhism and Jainism, Kailash is mainly represented as having been closely linked with a few very important people, such as Padmasambhava, Rishabhadeva and Milarepa. In some Tantric Buddhist sects, it is thought to be the home of the Buddha Demchok, who represents supreme ecstasy. The Bön faith, which is a lot older than Buddhism and was the original religion of Tibet, believes that the entire region, and this mountain in particular, are the source of all spiritual power and virtue on the Earth, and that this is where the originator of their tradition, Tönpa Shenrab Miwoche, was born.
One of the main religious practices for pilgrims, and one of the most important actions that Buddhists in particular can perform, is a circumambulation of Mount Kailash called the Kora. This can take various different forms, depending on your religion and your abilities or inclinations (or, to put it another way, how much pain you feel you can take). Hindus and Buddhists walk around Kailash clockwise, whereas Jains and Bönpos go anticlockwise. Ideally, the Kora is done in a single day, which is difficult since it's a 52km hike at high altitude and over tough terrain. However, some brave or foolish souls do something much more difficult: they circumnavigate the mountain using full-body prostration. This means that they have to kneel, then lie face down on the ground at full stretch, marking the fullest extent of the stretch with their fingertips. They then kneel again, pray, and crawl foward on hands and knees to the mark they have just made, followed by prostrating themselves again. It takes at least four days to go around Kailash in this way, but if you manage it, presumably you have enough spiritual credit to last you the rest of your life (although some truly hardcore traditions state that it's better to do it 3 times or more, with one doctrine holding that if you do it 108 times you go instantly to Nirvana without passing Go).
Following the Chinese occupation of Tibet, pilgrimages to Mount Kailash were stopped from 1959 to 1980, and since then have begun again, under control of the Chinese government, which grants a certain number of permits per year to Indian pilgrims, or to its own (Tibetan) people. The Chinese government doesn't feel too warm and fuzzy about religious traditions, as is well-known, and has experimented in the past with granting climbing permits for Mount Kailash, but they changed their minds after extremely heated protests by adherents of the affected religions, particularly Tibetans.
Rumours have persisted about the mountain having been secretly summited, but there is no evidence that this has actually happened. One of the rumours is that Reinhold Messner climbed it, alone, and at night, and was back down off it again by the time light returned; this is extremely unlikely. The rumours notwithstanding, no one has ever admitted to climbing or attempting to climb Mount Kailash, as it would make them both a target of religious outrage and outcasts of the international climbing community.
Kora travel blog: http://www.travelblog.org/Asia/China/Tibet/blog-29907.html