REINHOLD MESSNER

He was a great rock climber, until he lost some toes in the Nanga Parbat epic where his brother died.
And he became, without any doubt, the greatest himalayist of all time

Just some well known facts:

1)First climber to complete the 14 8000m peaks

2)First solo of an 8000 peak (Nanga Parbat)

3)First ascent of Everest without oxigen (with Habeler)

4)First solo ascent of Everest

5)And he is still ALIVE !!!!!
(unlike Kukuckza, who died on the south face of Lhotse)

Lowe - nuff said
Messner - all time role model
Humar - most underrecognized climber, at least in the states

I love how the OP asks for "best all around climber including mountaineering" and gets names like Graham, Hill, etc.