The typical conception of LAMP doesn't include desktop machines, it is about a web-based application stack. I suspect there are very few people using LAMP to solve every business need they have. Partly, this is because there are very few businesses that can (or, at least, should) address all of their business needs with a web-based application stack of any kind.

There are many successful businesses that use LAMP as their primary, or indeed only, web application stack. There are even a few businesses that use Free Software to solve all of thier business challenges. Most of these aren't very big. Consider why:

1. Conversion to pure-Linux (for example) on the Desktop is extremely costly. Starting a business or business unit with Linux is far less costly than converting.
2. Linux has only relatively recently become viable for desktop deployment to average users, and even more recently been accepted in some business circles as a Windows alternative.
3. Technology lock-in is typically a bad thing. While the open nature of Linux and most distributions of it make it less of a problem, most wise business owners don't like to put all their eggs in one basket unless they have no other choice.

Because of these, it's mostly relatively new entrants that are using Free Software extensively in every aspect of their business. That usually means smaller, lesser-known entities.

However, I can tell you that many governments are exclusively Linux on the desktop, and are moving away from Windows in the server space as well. Take the Minnesota DOT, for example.