There seemed to be several solutions to this problem and I even came up with one of my own.
Similar to @iglvzx I too used the AutoHotKey utility to create my own utility app.
I've posted the app online here: http://dl.dropbox.com/u/177276/ipAddress.exe
I throw my exe into my Startup folder... and it sits quietly waiting until I hit the hotkey:
WindowsKey+I
which then brings up this dialog... allowing me to copy the address with a single click... or navigate to another window as the IP address will display on the app tab on the start bar. I ended up opting out of automatically putting it on the clipboard just in case I had something important on there that I didn't want to accidentally delete.
Here's the source code I used to get the IP address (it makes a presumption that the %A_IPAddress1% is the correct one (but from my testing it always was)):
#SingleInstance#PersistentMenu, tray, NoStandardMenu, tray, add, Exit, ExitAppCompletelyHotkey, #i, ShowIPAddressreturnShowIPAddress:Gui, Add, Text, x50 y8, Your IP Address:Gui, Add, Edit, x140 y5 ReadOnly vIPAddress, %A_IPAddress1%Gui, Add, Text, x50 y35 w250 vCopiedStatus,Gui, Add, Button, x70 y65 w75, &CopyGui, Add, Button, x150 y65 w75, &DismissGui, Show, W290 H100 Center, %A_IPAddress1% - IP AddressreturnButtonCopy:clipboard = %A_IPAddress1%GuiControl,, CopiedStatus, Copied %A_IPAddress1% to the clipboardSleep, 1000GuiControl,, CopiedStatus,Sleep, 500ButtonDismiss:GuiClose:Gui, DestroyExitExitAppCompletely:ExitApp