We've Already Had Some...
...the most noteworthy was Dennis Tito, who spent about a week on the International Space Station (ISS) a couple of years ago. He paid the Russians 10's of millions of dollars for the training, a ride in a Soyuz spacecraft and the ISS visit. It really ticked off NASA, BTW. The Russian had originally contracted with Tito for a visit to the Mir. By the time they had him trained and scheduled, the Mir had been deorbited. The Russians "informed" NASA that they would be bringing Tito to the ISS in order to fulfill their contract. NASA put up a fight that caused several month's delay in the visit, but they eventually agreed with a whole bunch of constraints -- like "don't touch anything!" NASA was really two-faced about the whole subject. In the pre-Challenger days, I recall they were lining up all sorts of people (including politicians like Jake Garn & Bill Nelson, school teachers, Walter Cronkite, etc) for Shuttle rides who were, at best, marginal contributers to the space program. (I have to exclude John Glenn, who had a key medical experimentation role on his Shuttle flight. On the other hand, Jake Garn's job was to be the case study for space sickness -- the designated puker -- a role I'm told he performed very well and very often.)
In addition to the money aspect, you can't be over about 5'6" in height and fit into a Russian Soyuz spacecraft. I'm 6'0" tall and I've tried to fit into one at RSC Energia -- no go!