In Sweden, you can have three, four or more names without having a middle name. A middle name is a surname, either the mother's or the father's. You have to tell the tax authorities that this is a middle name. Then the child is officially call yourself "Ylva Jansson Lind" for instance. One parent's surname is Janson, the other's Lind. If the mother has taken the father's surname, the children cannot not have a middle name.
Then we have "tilltalsnamn" (the name people call us, don't know what the English word is). This we also tell the tax authorities. It is usually the first given name, but can also be the second or the third. YlvaElisabethMaria may be called Ylva, Elisabeth or Maria.
People sometimes say that we should place the "tilltalsnamn" first, b/c even if it works in Sweden, it does not work in USA. Of course, most of us don't move to USA, so I don't think this is a valid argument. I think the rhytm of the names is more important.