10. The Company gets lazy by the time they reach Moria and decide to simply toss the ring over into the abyss beneath the bridge at Khazad-dum.
9. When the Hobbits are first overtaken on the road by a Nazgul, Sam gives away their position by trying to "peg that black chap with an apple".
8. Boromir manages to take the ring, but it turns out that all that stuff about the ring causing owners with good intentions to do evil was just a buncha crap.
7. Sauron himself comes to the Shire to claim the ring, but after a few bowls of pipe-weed and a few pints of beer decides to "give up all that evil and enslaving junk" and settle down in the Shire.
6. Instead of Sam, the Gaffer accompanies Frodo, but his incessant name-calling and long rambling stories about the origins of "taters" causes him to be cast into the cracks of doom along with the One Ring.
5. Tom Bombadil pulls his disappearing act with the ring, but can't seem to remember how to make it reappear.
4. Tom Bombadil claims the Ring and has all the denizens of Middle-Earth picking water lilies and singing nonsense off the top of their heads.
3. Gollum claims the Ring and now sits in Barad-Dûr, where he eats fish all day, and has made his gibberish the official language of Middle-Earth, precious (gollum).
2. Seizing the Ring, Samwise Gamgee puts all of Middle-Earth onto huge farm collectives, ala communist Russia.
And the number one Rejected Plot Change is...

Seeing that his Ring has been destroyed, Sauron still manages to take over Middle-Earth with a marketing blitz and several hostile takeovers: Red Eye brand Lembas, Prince Sauron Pipe-Weed, and genuine troll made Dwarf jelly, available also in Hobbit, and soon Elf flavors.