What is the difference between 一つ目 and 一番 and 一番目?
As far as I know they all mean first, but I don't know what the difference is. What is the 目 tagged onto 一番, and how does it change the meaning, and how is it different to 一つ目?

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I think I've also seen 一番目 in the context of first line(of text), first sibling, so it's a first of some order/list. 一番 on the otherhand often is used to mean "the most". 一つ目 I've never heard of.. I hope this helps. :)

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*If they are out of order.
*目 or 番目 was ...
Excuse me :(

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1 is 一番, and then 一番目 is the first number among numbers.
For example, there are numbers from 1 to 5 in order like 1 2 3 4 5.
You can tell that 1番目 is １.

Then, how about these numbers? like 3 2 5 1 4. 
If they are out of numbers, you can say 一番目 is ３.

目 or 番目 were created for 歌舞伎 several hundred years ago, and that is the difference between 一つ目 and 1番目.
Also, you may use 一つ目 for only a thing, while you can use 1番目 for both a thing and a person.

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目means a thing/matter maybe, and it is used when you choose one thing from something a lot. For example, who was your first boyfriend? "一人目の彼氏は誰ですか？" Sorry for my bad explanation:/