865
But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
They should have laid a long garden hose all the way to mt doom and pushed the ring though it with a bicycle pump
You joke but the disregard for physics in the lord of the rings trilogy was astounding; giving an item to another person is a free action, thus if you create a line of people all the way to mt doom, you can pass the ring all the way from the shire to mt doom in six seconds at the speed of light; heck if the last person in the line just throws the ring at the eye of Sauron it would be like a kinetic weapon with the destructive force of the deathstar.