for example, 'for another example, for the third example.... or just one 'for example'?
during composition, I give three examples. the first, I begin with ' for example' to introduce my example. then I want to give the second example. In this situation, I should write the second example directly following the first one, or I need to write ' for another example' to remind the readers another example is coming? (I doubt its correctness because I have never met it)

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It depends on your composition... If you were just listing two or three short items in a single sentence, you could say "For example, ,  or ." If you were doing much longer examples or were separating them into paragraphs, then you would need to reintroduce each one just to keep the focus: "For example, . A second example is "

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try not to become monotonous
For example................................ Furthermore........................................, and again....................................................

fill in the blanks with your examples. But yes, you can call those 2nd example, 3rd example etc

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I will give three example:

1. ...

2. ...

3. ...