If we say "God the Father, God the Son, and God the Holy Spirit" can we say "God, God, and God"? God is said to be "the triune god". He exists as "the Father" (who is God), "the Son" (who is God), and "the Holy Spirit" (who is God). 

Since Jesus is "the son of God", then the Father is independently referred to as God. Otherwise, Jesus would be the son of the triune god, and he would begat himself. 

Because of Communicatio idiomatum, Jesus is also referred to as God. So it is said "God died for our sins" or "God suffered". 

So we have the Father who is referred to as "God", the Son who is refered to as "God", and there is also the Holy Spirit guy, who is likewise referred to as "God".

Is it correct to refer to God as "god, god, and god"?
Language uses words with particular senses to talk about the world. It's a system of symbols which point to things in the world. One thing in the world, one referent, might have many different words (with many different senses) which can be used to refer to it. For example, one person could be referred to with the words "Mrs Smith", "Ruth", "mother", "grandma", "boss" etc. Though all of those words refer to the same person, they do not have the same senses and are not interchangeable.

Here's another example: at a meeting of world leaders the President of the United States, the President of Russia, and the President of Mongolia all had lunch together. In the context of each country's own media, it is common and appropriate to call each by simple title of "the President". But it would be very confusing to transform that statement into "at a meeting of world leaders the President, the President, and the President all had lunch together." And that's not just because they're all the same word; it would be wrong for US media to write "the President and the Chancellor of Germany had lunch together" if they were referring to the Russian president.

More is going on here than just confusing language however. The unqualified word "God" in Western Christianity (Catholicism and Protestanism) usually refers primarily to the Godhead, the singular essence of God, the one divine being in existence. My understanding is that this is one of the differences between Western and Eastern Christianity: instead of using "God" to refer to the godhead, Eastern Christianity usually uses it to refer to God the Father specifically.

When we say "Jesus the Son of God" we are referring to the second person of the trinity in particular. When we say "Jesus is God" we mean something different, that he is divine. But this does not mean that Jesus is the Godhead or the divine essence.

So if someone said "God, god, and god" Western Christians will only hear the first sense, that they are referring to the godhead three times, which means three divine beings, three divine essences, which is tritheism. Eastern Christians might instead understand it as referring to God the Father three times. But in neither case will it be understood as referring to the three persons of the Trinity.