This is a contrived example of what I want to do, but minimally expresses the behavior desired. I want to reference the instance of the object on which the property access is being invoked. I tried 'this' first, but that refers to the enclosing class rather than either the MetaClass or the String instance.
String.metaClass.propertyMissing = { String name ->
'I do not exist, but my name is ' + <the String instance> + '.' + $name
}