Strangely enough, I was just reading a newspaper article about this the other day. There are adventure travel agencies which, for the right price, will organise expeditions to Everest. There's a false impression that they'll take anyone who can come up with the money, which (at least for the reputable ones) isn't true - you still need to prove to them that you have the skill, experience, and fitness to participate. (the article didn't list any of these agencies, so you're on your own there - but there are lots of travel agents who specialise in adventure tours, they'd be a good starting point.)

A friend of my sister's and her fiance went on an expedition to the Everest base camp some years ago. Similar deal, arranged through a package outfitter. They never made it to the base camp - after spending several thousand dollars, the fiance came down with a case of the sniffles after a few days on the trail and the agency medevaced him out (she chose to go with him) because they couldn't take the risk of it being altitude sickness. Apparently they just shrugged this off - although disappointed, they felt that it was the price you paid for dealing with a reputable agency that placed their customers' safety above profit. (No, they didn't get their money back.)