1683 - November 15, 1706: Age 23
Tsangyang Gyatso was one of those men you can't put in a box, figuratively speaking. He was recognized at the age of 5 as the reincarnation of the Fifth Dalai Lama, but immediately put into hiding. The political situation in Tibet was deemed too unstable to announce the death of the old Dalai Lama, and in medieval Tibet it was not impossible to hide someone's death from much of the country and the outside world for some years. In 1697 the death of the Fifth was announced, as was the news that the Sixth had been found. Thus began Tsangyang Gyatso's short career as a public figure.
In 1701 the Regent, who had been protecting him, was the victim of a Mongol-ordered assassination (Tibet, China, and the Mongols have a long history of mutual influence and interference). The young Dalai Lama left his studies and renounced his monk vows, and embarked on a life of booze and romance. He continued to function as Dalai Lama, but insisted on wearing layman's clothes and refused to ride in a special palanquin and other spiritual perks. The love songs he wrote for his paramours are very highly regarded in Tibetan literature.
Using this behaviour as an excuse, in 1706 a Mongol king deposed Tsangyang Gyatso and declared a 25-year-old Lama the "real" Dalai Lama. The Tibetans would not accept him. Another Mongol tribe was invited in to oust the one that had deposed him — this is classic Tibetan history, there's always another Mongol tribe to invite in for an invasion. By 1717 the unwanted Mongols had been defeated, but it was way too late — Tsangyang had died mysteriously in 1706 while being spirited out of the country.
Sources: Wikipedia, BBC News
Note: I can't resist pointing out that at around the time Tsangyang was being kidnapped, Johann Sebastian Bach was walking 400 miles to Lübeck to study organ with Dietrich Buxtehude. Music nerd stuff.