Greatest among any other town is impossible. That is a comparison with only one other town, and one town (or population) cannot be "among" only one other.

Your first suggestion, greater than any other town, is acceptable, but a little loose; your second is even better, because it compares populations instead of comparing a population to a town: it answers the question "Which of these two populations is greater?"

Those constructions compare Tokyo's population to that of each other town, one by one. You may employ the superlative greatest only if you compare Tokyo's population to those of all the towns in a group of at least three towns. When performing a comparison of this sort, Tokyo itself must be included in the group whose populations are compared, so you may not use the term other.

This is a little trickier to construct with precision, because when you refer to the multiple populations you must be careful to exclude the possibility that you are comparing Tokyo's population to the sum of the populations of the other towns. You may accomplish this by "detaching" each town from its population and asking a slightly different question: "Which town has the greatest population?"

Among all towns in the world, Tokyo has the greatest population.
Tokyo has the greatest population of any town in the world.

The quoted sentence is not the way a native English speaker would say it, as "greatest" is the superlative form of great - this means that if something is "greatest" then it is the same as saying "greater than any other". As such, the phrase "...than any other" or "...among any other" is to be avoided. One unusual exception is in the phrase "greatest of all", which is a special case of a superlative phrase and can't be generalized to all uses of greatest.

As to your two proposed alternatives, they are both superior to the original and are perfectly acceptable; however, I should warn you that "great" has multiple meanings that could be misinterpreted in some contexts (it can refer to quality or to number). This is the subtle difference with your two proposed sentences, as the last prevents misinterpretation with "greater than that". This is a relatively minor consideration as most speakers understand when 'great' is being used qualitatively or quantitatively - but the danger of being misunderstood is greatest when you are referring to types or sets of people.