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 mean that's the reason Hobbits in general can withstand the Ring longer than any other race of Middle Earth. They just want a quiet life without any fuss and that's pretty much the opposite of what the Ring can promise them.
Ring: I can make you rich!
Hobbit: Eh, than my cousins will pester me all day.
Ring: I can make you strong!
Hobbit: What for? I have an ox for that.
Ring: I can make you king of all!
Hobbit: That's even worse than rich!
Ring: Exasperated sigh
Ring: Fine! I can give you third breakfast!
But I'm busy eating elevenses right now! Maybe we can have third breakfast tomorrow?