Although the 2nd highest mountain, K2 provides a ridiculously challenging technical difficulty namely due to the infamous bottleneck and shoulder before the summit. This provides a level of objective danger not really matched elsewhere.
Due to its Northern location in the Karakoram the weather is notoriously difficult. Climbing it in the winter is one of the last great challenges in mountaineering.
It's well worth reading up on its incredible history. Particularly the Duke of Abruzzi who actually got very close for the day and age. Also Alestair Crowley's attempt in about 1902.