As said before, it really depends on the definition of greatest. By strength, skill, battle prowess, and achievements, the greatest of the races certainly have been already named.
However, we probably have to see the whole thing within the cosmology, which is unevitably tied to the morals and values of Middle-Earth, in order to give a judgement whom the author would have assumed as being the 'greatest'. For some, he said it more or less explicitally in the books - such as Fëanor being the greatest elf in skill and mind alike (not verbatim, but close, I think).
Nevertheless 'my' 'author's choices' would be something like that:
Lúthien of Elves, (can't get better than 50% Maiarin) or maybe Ingwë or Finwë.
Eärendil of Men. He was much greater in the myth than the short fragment of the Silmarillion tells. Other choices might be Aelfwine or Eriol.
Dwarves: Durin the Deathless. Interestingly, the opinions diverge over this practically quite clear case.
Hobbits: Bilbo Baggins or Samwise Gamgee. Tough call.
Balrogs and orcs? There were none that were 'great'. If you mean 'mighty (in evil', Gothmog is the obvious one, and Boldog for orcs, I think (although Uglúk was a meanie, too).
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