A fascinating question and one that's solved by exploring the Profile Manager that's included as part of Firefox.

To start up Profile Manager in Windows, close Firefox by selecting File --> Exit, then choose Start --> Run... from the Windows START menu. Type in firefox.exe -profilemanager and click OK.

On a Linux systems, launch a command terminal and type in firefox -profilemanager.

On a Mac, well, on a Mac it's a bit more of a hassle because of how Mac OS X hides the actual application binaries. Let me show you how I got it to work on my own Mac OS X system...

First off, launch Terminal (Applications --> Utilities --> Terminal), then move into the directory where Firefox is located (probably /Applications/Firefox) with the command cd /Applications/Firefox. Now you have to go a bit deeper to get to the correct place, so enter another command: cd Firefox.app/Contents/MacOS. Now, finally, you can enter the necessary command. Here's all three, to make things a bit clearer:

Here's what you'll see:

The key setting you want has already been checked in this screenshot: Don't ask at startup.

Check that and you'll find that you won't be prompted to select a profile when you start the Firefox browser in the future.

A simple solution, though it'd be nice if Firefox itself could launch the Profile Manager instead of making you jump through all these hoops!