Hi Galik, greatest has the specific meaning of top of his class. Major in this case would mean very important or one of the greatest and it would require an a (X is a major naval hero in British history)
The way the sentence is written may have got you confused. I understand it as:
Nelson is Britain's greatest naval hero.
As you can see a "the" before greatest would be odd here.
Saying that Nelson is a major naval hero in the history of Britain, would be a huge understatement
It would not be ungrammatical to sayNelson's, but it is unusual to have two words together like this, especially where they are different constructions (e.g. Nelson's = Nelson is; Britain's = of Britain).
I think the main reason is one of sound. You might say My car's exhaust's broken, where the 's is similar in both cases. But you would not write it.