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
I forget the exact wording, but the Ring essentially showed Sam visions of being some sort of a supreme gardener king. Sam dismissed that as fucking stupid, because he just wants a simple garden.