Note that the convention of what is your given name and what is your family name, and how they are inherited are different that the convention in USA. This add a lot of confusion when someone moves and need to fill paperwork. Using a "|" to separate your given name(s) and your family name(s):

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Common pattern in USA:

Father: X Y | Z

Mother: A B | C

Your name: R C | Z

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Spain/most of Latin America:

Father: X | Y Z

Mother: A | B C

Your name: R | Y B

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Argentina:

Father: X Y | Z

Mother: A B | C

Your name: R S | Z

(Catholic?) Germany:

Father: X | Z

Mother: A | C

Set of Godfathers and Godmothers given names: {G}

Your name: {G} | Z

There are also double given names. For example Hans-Peter is a differs from Hans Peter. The later could also be rendered Peter Hans but for Hans-Peter the order is fixed.

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Many Slavic Countries:

Father: X | Z

Mother: A | C

Your name (boy): R | patronymic(X) | Z

Your name (girl): R | patronymic(X) | feminine(Z)

The middle part is neither given name nor surname but referred to as patronymic.

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> Try explaining you have three to a clerk...

Or even that you don't have a first name but a set of given names and that the relative that gave you the name that is conventionally often put in a later position would not be amused about another relatives name singled out to be a "first name".

Is it harder than just saying "I have three middle names"? It's not exactly a foreign concept; it is normal for white Americans to have more than one, though not especially common.

Three "middle names" might not be a problem but the lack of a first name would almost certainly lead to troubles. Of course you can always just choose one of your "middle names" as an artificial "first name". Some people might find this disrespectful though.