854
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
The animated films had a segment of when Sam had the ring and was going to rescue Frodo, and it tempted him that he could use its power to set the world right. I kind of wish Peter Jackson would've done something similar.
https://youtu.be/1PE-5ETUtW4