What is the average big brand spend on a cool website?

It really depends on a lot of factors. They will hire an agency to build the website, but it all depends on what they're getting from the agency. If they're having the agency maintain the site over time and continuously optimize it, it's going to cost a lot. If it's just to build a site that and they know exactly what they want, it will probably cost at least a few hundred thousand. The big brands, however, will likely hire an agency to work on the strategy of the site, the build of the site, and the maintenance of the site which will cost a few million depending on the specifics of the agreement.