866
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
Sam did bear the weight of the ring, its hard to convey in the movie but the book makes it clear. Sam just had an iron will.
AND he hadn't already been carrying the ring the whole fucking way like Frodo did!