864
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
This would create quite an evil mouse. I imagine the risk here is that the mouse would break free and run away with the ring and bring it to Sauron.
Edit: Someone already beat me to it.
You need legs to run away
Also good luck finding an invisible mouse.
it would take the cookie, and you know what happens when you give a mouse a cookie