Can the banner in the Home Edition be removed?

The process for Mathematica 9 is different. Home Edition 9 appears to lock out certain options in the Option Inspector, including the ones required to make this change. It will appear to permit the previous change, but will immediately revert.

To make the change for 9, you need to open up the MiscExpressions.tr file (on Mac it's in the Mathematica.app bundle, under /Applications/Mathematica.app/Contents/SystemFiles/FrontEnd/TextResources/MiscExpressions.tr.)

Then find the definition for HomeEditionBar, the first line looks like this:

"HomeEditionBar" -> Cell[BoxData[GridBox[{

Change that line to these two:

"HomeEditionBar" -> Cell[],
"HomeEditionBarOriginal" -> Cell[BoxData[GridBox[{

And now you have the full height of your screen to work with.

Caveat: This is likely prohibited by the EULA, which does not permit modification of any part of the software.

OK, something better, without editing files manually. No EULA violation whatsoever.

Select Option Inspector in the Format menu:

and change WindowFrame in Window Properties to Generic. Restart Mathematica and the banner has gone. (Actually, all the options for WindowFrame except Normal will remove the banner, but they will also change other aspects of the window's look.)

Edit
I first suggested an edit of a particular init file here, a method I found here. Both my question and this answer caused quite a fuss about the legitimacy of it. In hindsight I don't think editing init files should be the appropriate way (which does not mean it would be illegal), especially since Mathematica allows you to make the same changes Rob makes. Clear the user option field DockedCells in the Option Inspector: