I assume there are probably practical limits to how high an elevator can be, based on - if nothing else - the sheer weight of the cables. The elevator will probably have to ascend in multiple stages.
Staging!
Well, there are 5 camps.
I couldn’t listen to The Girl from Ipanema for that long.
I don’t see how it could be an external elevator because of the weather and the snow/rock slides. To bore a shaft from the base all the way up to the summit seems like a fool’s errand to me. You’ve got to construct an elevator system that goes up the equivalent of almost 3,000 floors.
You know, as long as we’re hollowing out sections of the mountain to build an elevator, why not hollow out a lot of the mountain, and make it an arcology?
Who wouldn’t want to have their permanent address as, Mount Everest? 
Several hundred people already do.
10 hours of the Girl From Ipanema:
Sure but we already have an elevator that can go up 140 floors in a minute. With a full restaurant you would prefer to have people at various stages of coming and going so build 22 stages and it takes half an hour to travel the full distance, it would be part of the experience. Each floor could have music, a bar and snacks. The elevator isn’t the hard part of this idea.
Each floor can have music, bar snacks AND TOILETS,
~VOW