Jon Krakauer is, in my opinion, the very best nonfiction outdoors writer, all of the following are highly recommended:
Into the Wild
Into Thin Air
Eiger Dreams
Others that get a full fledged two thumbs up:
Wild: From Lost to Found on the Pacific Crest Trail (by Cheryl Strayed) - Actually, I like her writing as much as I like Krakauer's, which really is saying a lot.
Minus 148 degrees: The First Winter Ascent of Mount McKinley (by Art Davidson)
A Walk in the Woods (by Bill Bryson) - really pretty funny
Touching the Void (by Joe Simpson)
Here are also some very good instructional books, depending on one's aspirations:
Allen and Mike's Really Cool Backcountry Ski Book, Revised and Even Better!: Traveling and Camping Skills for a Winter Environment (by Allen O'Bannon and Mike Clelland)
Glacier Mountaineering: An Illustrated Guide to Glacier Travel and Crevasse Rescue (by Andy Tyson and Mike Clelland)
Glacier Travel and Crevasse Rescue (by Andy Selters)
How to Rock Climb (by John Long)
Climbing Anchors, 2nd Eddition (by John Long)
Performance Rock Climbing (by Dale Goddard and Udo Neumann) - I like this one because it addresses the topic of fear, which can apply to other persuits as well.
And then there's
this list, which is from Outside magazine. Several in there are on my short list, but I have not yet read.
Enjoy!