The correct option is
D others
The sentence talks about two kinds of people-those who are born great and those who achieve greatness. There is a difference between the two set of people. The words, 'some... others' are used in the same sentence to contrast two different thoughts/ ideas. The sentence, 'Some are born great, others achieve greatness' conveys: 'Some are by birth, great; However, others become great in their lives. Hence, option D is correct.
Option A: Repeating the same adjective, 'some' multiple times over in a single sentence is not agreeable in English. It also does not convey the expected contrast. Hence, this is incorrect.
Option B: The word, 'any' is used to refer to one or some part of a thing or number of things. It conveys an ambiguous meaning and does not express the required contrast. It is also an adjective and needs a noun to complete its meaning. Hence, this is incorrect.
Option C: The sentence, 'Some are born great, no achieve greatness' is grammatically incorrect as 'no' is usually a determiner which needs a noun to be used along with it. Hence, this option is incorrect.