It isn't easy, partly because you have to work with the Selection because
you can't get the Range of a Field using e.g. oField.Range.

You might find the following article by Dave Rado at the Word MVPs site
useful (you should also note that dealing with fields in Headers/Footers may
be even harder):

http://word.mvps.org/faqs/macrosvba/NestedFieldsWithVBA.htm

I don't have any tested general-purpose code that deals with nested fields.
I threw the following together as a starting point - you would need to test
the approach. I expect a more experienced Word/VBA coder could do much
better!

Sub ListFieldCodesInMainDocument()
Dim bIsOutermostRange As Boolean
 bIsOutermostRange = True
 Call ListInnerFields(ActiveDocument.Range.Fields, bIsOutermostRange)
End Sub

Sub ListInnerFields(oOuterFields As Word.Fields, bIsOutermostRange As
Boolean)
Dim oInnerField As Word.Field
Dim oInnerFields As Word.Fields
Dim bFirstFieldInRange As Boolean
' The way I have done things, we need to eliminate the first field in the
collection
' unless we are dealing with the outermost collection
bFirstFieldInRange = True
For Each oInnerField In oOuterFields
 If bFirstFieldInRange And Not bIsOutermostRange Then
   bFirstFieldInRange = False
 Else
   oInnerField.Select
   If Selection.Range.Fields.Count > 1 Then
   ' nested field
   ' do something with the outer field
     Debug.Print oInnerField.Code
     Set oInnerFields = Selection.Range.Fields
     Call ListInnerFields(oInnerFields, False)
     Set oInnerFields = Nothing
   Else
     ' do something with the inner field
     Debug.Print oInnerField.Code
   End If
 End If
Next

End Sub