Shall I write: Three OTHER examples or Three ANOTHER examples???

thank you
But the sentence is not saying that. It is saying there are three other [= alternative] examples.

By saying another, you would be saying "Here is an example, here is another example". Another is used as one of two possibles >> "It is one or another of the examples".
I don't know if this is how the word was formed but it might be useful to think of it as 'an-other'. 'An' can only apply to one thing so think of this as meaning 'one more' or 'one other'.

Would you like another cake? - You are only being offered one more cake.

Can you find another example? - You are only being asked for one more example.
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Three other examples
Feebs11Three other examples

I would say: One example, the other example and another example.

That's why it seemed quite reasonable for me to say: Three another examples rather than other.
 Feebs11's reply was promoted to an answer.
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Thanks. I'm not sure I get it, but thanks.
NewguestThanks. I'm not sure I get it, but thanks.

Feebs gave you a clue: "Another is used as one of two possibles"

You can use another for more than one thing if you group them in group. Here is what I mean:

We can say, "three students finished their homework, another three [students] did not." (not 'three another students did not')
 nona the brit's reply was promoted to an answer.
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What the heck are you talking about?! Just write as many examples you can think of that are good. It will really help all the people that are confused with metaphors and need help to think of one.