Ok, here I go, I'll try not to re-iterate too much...
It all really depends on how you measure said strength. Plus, you haven't factored in the other, smaller races like the Woses, Ents, Eagles or Dragons.
In the game: This usually comes down to elves, or dwarves. Sure, the elves are better in almost
every way (which is wrong IMO) but with dwarves, you'll have a force that is rock-hard and fairly skilled with most weapons, albeit being kept quite tightly bunched up. However, an elf or dwarf (although dwarves can be quite cheap) army is almost always expensive (in points and money) whereas other races, especially hobbits, cost a lot less in points, and will have less numbers difference between them and an evil army.
IMHO it's between men and dwarves; men are VERY balanced, all-rounded armies, they're also cheap, with some amazing heroes to choose from. Dwarves, whilst they have no cavalry, are quite strong- high defence, *reasonably* high fight value, courage et cetera.
In the books: Affirmative. they were obviously at their strongest in the first age, weathering attacks from Sauron AND Morgoth together
and still able to give about as good as they got. Even well into the third age they were strong.
... However, I would like to make one minor point: The dwarves were pretty much equal to the elves in craftsmanship-particularly the area of weaponry; eg Angrist, mithril armour, the horn which Meriadoc got from Eowyn, Narsil (yes, elves did reforge Narsil into Anduril).
@aelfwine:
The elves are not demi-gods, that's what The Maiar are.
@Thorin and co. : The dwarf kings became selfish, greedy and secretive after receiving their rings....
*wipes forehead*