Which are the most expensive LEGO sets?
The most expensive (and second largest) LEGO set to date is the #10179 LEGO Star Wars Millennium Falcon released in 2007, with 5,195 pieces, and the cost of $500. Equally expensive (also $500), but much smaller is the recently released #75159 LEGO Star Wars Death Star with 4,016 pieces. 

As far as what is the most someone paid for a LEGO set in a private transaction would be impossible to tell. You could use the BrickLink Price Guide for prices sets sold for in the last six months, and you can also do something similar for eBay, but there are many other places people buy and sell LEGO (social-media sites, forums, commercial websites, auction sites, and more). You would really have to narrow down what data you are looking for, and go from there.