OK, so now I'm getting the strange behaviour that I'm being asked if I
want to save twice.

Here's what I've got:

in module EventHandlers:
-----
Public wehEventSink As New WordEventHandler
-----
Public Sub AutoExec()
   ' Register the Event Handler
   Set wehEventSink.WordApp = Word.Application
End Sub
-----

in module class WordEventHandler:
-----
Public WithEvents WordApp As Word.Application
-----
Private Sub WordApp_DocumentOpen(ByVal docOpened As Word.Document)
   MsgBox "WordApp_DocumentOpen event handler"
End Sub         ' WordApp_DocumentOpen
-----
Private Sub WordApp_DocumentBeforeClose(ByVal docClosing As
Word.Document, ByRef Cancel As Boolean)
   MsgBox "WordApp_DocumentBeforeClose event handler"
   If Not ActiveDocument.Saved Then
   Select Case MsgBox("Do you want me to save the changes to " & _
   Split(ActiveDocument.Name, ".")(0) & "?", vbYesNoCancel +
vbExclamation, "save file")
   Case vbYes
       MsgBox "User opted to save. Saving and quitting"
       ActiveDocument.Save
       Call GetFileSize
   Case vbNo
       MsgBox "User opted not to save. Quitting without saving"
       ActiveDocument.Saved = True
   Case vbCancel
       MsgBox "User opted to cancel. Cancelling quit"
       Cancel = True
   End Select
End If
End Sub
-----
Private Sub WordApp_Quit()
   MsgBox "WordApp_Quit"
End Sub
-----

in module FileSize:
-----
Sub GetFileSize()
MsgBox ("GetFileSize()")
Dim OpenFileSize As Long
Dim OpenNumPages As Long
Dim NewFileSize As Long
' Get Filesize of saved document
OpenFileSize = ActiveDocument.BuiltInDocumentProperties("Number of
Bytes")
' Get Number of pages of document
OpenNumPages = ActiveDocument.BuiltInDocumentProperties("Number of
Pages")
' Compare
If OpenFileSize / OpenNumPages > 30000 Then
   Dim intMsgBoxResult As Integer
   intMsgBoxResult = MsgBox("The file is only " & OpenNumPages & "
pages and currently over " & FormatNumber(OpenFileSize / 1048576, 2) &
_
       " MB.  Would you like to compress images?", vbYesNo +
vbQuestion, "Bloated File Size")

   If intMsgBoxResult = vbYes Then
       'Do something
       Call CompressPics
       ActiveDocument.Save
       NewFileSize = ActiveDocument.BuiltInDocumentProperties("Number
of Bytes")
       MsgBox ("The new file size is " & FormatNumber(NewFileSize /
1048576, 2) & " Mb. " & vbCr & vbCr & "That's " &
FormatNumber((NewFileSize / OpenFileSize) * 100, 0) & "% of the
original.")
   End If
End If
End Sub
-----

When I make changes then quit the file, I get the message boxes
indicating the sequence:
  DocumentBeforeClose
  "Want Me to Save Changes?..."
     Y
  Saving and Quitting
  GetFileSize
  "Want Me to Save Changes?..."  <- here's the second one
    Y
  closes

It appears that the GetFileSize routine touches the file (probably the
OpenFileSize and OpenNumPages), and changes the Save status.  So,
before the file *actually* closes, it brings up the custom FileSave
dialog again.

Any ideas?

Thanks!