Read up on TANSTAAFL[1]. Sure, you may not be paying anything for software licenses, but you will still have costs for installing and maintaining the software, which for certain scenarios will far outweigh the licensing costs. So you have to look at the <buzzword alert>TCO</buzzword> of a given software deployment scenario before you can make a call whether a setup consisting exclusively of open source software is (more) profitable for you.

As for your original question, whether such companies exist in the real world, I suggest you ask Jeff Bezos.

[1] I personally do not subscribe to the TANSTAAFL belief, because I think there are free lunches (and I prefere Iain M.Banks to Heinlein), but it's a good mental shortcut anyway.