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.
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.