It's not a "big company" per se but The Guggenheim is built on Joomla! It might be worth browsing the Joomla! Community Showcase to see what's in there.
You can get a list of sites running Joomla here - http://trends.builtwith.com/cms/Joomla! And download a list of almost 10,000 - http://trends.builtwith.com/websitelist/Joomla!
http://www.joomlacase.com/ http://community.joomla.org/showcase/ check this, you will get thousands of websites.
I'll used joomla before, the overall joomla is the best ever. Its also good for large company at all.
Yes, Joomla is good but just don't go for free extensions, instead good extensions. Whether free or commercial.
Not much when the ! doesn't get added to the URL ! http://trends.builtwith.com/cms/Joomla! That's better!