When you say "greatest", though, what exactly do you mean? The ones that took the most skill? The most courage? The most effort? The most determination? A combination of those things? Or the ones that contributed most to the "cause" of Middle-earth (loosely, I suppose, Eru's plan)?
If the latter, Gollum and Deagol's "achievements" would surely rate the lowest. And we should include Smeagol's part in the destruction of the Ring since, though not intentionally done, it nevertheless contributed to the "cause".
Taking a combination of those things, but bearing in mind the morality of the tale, I would rate them as follows (with some new ones added):
1-Frodo brings about the destruction of the Ring -
See comments above.
2-Merry helps slay the Witchking of Angmar -
Immense courage, overcame great fear and an important contribution.
3-Sam Gives the Ring back to Frodo -
It was not in Sam's nature to keep it, but it must nevertheless have taken an immense effort, being as he was in Mordor at the time.
4-Bilbo goes down the tunnel to first Smaug confrontation -
Rated higher than giving up the Ring as he did not know the nature of the Ring and was "assisted" by Gandalf, whereas knowingly facing a Dragon must have taken a lot of courage.
5-Frodo defies the Nazgul at the Ford of Bruinen -
No mean achievement given the shape that he was in.
6-Frodo decides to go to Mordor alone -
Would you have wanted to do that? Bormoir or no Boromir it was a selfless and courageous decision. Even moreso, in fact, given that he had witnessed first hand the Ring's effect on Boromir.
7-Bilbo leaves the Ring for Frodo -
See under 4. above.
8-Bilbo saves the dwarves from the giant spiders -
They were spiders, for goodness sake. And he was hardly a seasoned campaigner at this point.
9-Samwise rescues Frodo from Cirith Ungol -
Worthy of mention, and a high rating, I think.
10-Frodo agrees to take the Ring to Mordor -
I don't think he fully realised what he was getting himself into. But he knew something of the nature of the Ring, and he agreed to do so before he knew that the other eight members of the Fellowship would be accompanying him.
11-Pippin kills a troll outside the Iron Gates -
Killing a Troll is no mean feat for a young Hobbit. And he saved Gimli's life and nearly died in the effort.
12-Bilbo gives the Arkenstone to the elves and men of Dale -
He betrayed his companions. A difficult choice, but a wise one, and the right one.
13-Merry, Pippin and Sam organise the resistance to Sharkey's men -
They saved the Shire, and all on their own (leadership-wise) this time.
14-Lobelia confronts Sharkey's gang with her umbrella -
Quite a feat for an old lady armed only with an umbrella.
15-The Bullroarer oversees the rout a Goblin raiding party and beheads their Chieftain -
He ends up lower than I expected. But he had an army on his side, and Goblins just don't compare with Nazgul, Trolls, Dragons and Giant Spiders. And let's just forget the golf thing.
16-The Hobbit Archers join the struggle against Angmar -
A collective effort, rather than an individual performance, so not quite on a par with some of the others. But brave, nonetheless.
17-Bilbo gets the dwarves out of Thranduil's Realm -
Not an inherently dangerous task, but one which showed resourcefulness and involved a great deal of discomfort.
18-Fatty Bolger stays behind to cover Frodo's trail -
As has been said, he didn't realise the implications of what he had agreed to do. It would have been braver to go with the others. But raising the alarm was a smart move.
19-Bilbo enters a riddle contest with Gollum in the darkness of the Misty Mountains -
I would rate it higher, but he did cheat.
20-Pippin looks into the Palantir -
He was a fool of a Took to do it, but it had important consequences.
21-Gollum falls into the Crack of Doom with the Ring -
Not intentionally done, but it saved Middle-earth.
22-Bilbo returns to Erebor after giving up the Arkenstone -
The deed was already done. The courage was in taking the Arkenstone. Once he did that, he had to face the consequences.
23-Rosie has numerous children and emotionally supports Samwise -
Impressive. But hardly extraordinary. Sorry.
24-Gollum's epic efforts to regain the Ring -
OK, so it showed great determination and resourcefulness. But it was selfishly motivated, and I cannot really credit that.
25-Deagol finds the Ring in the Anduin -
An act that had far-reaching consequences. But it hardly involved much input on Deagol's part. The Ring wanted to be found, anyway.
I am sure that there are more that could be added - there must be other Hobbit deeds greater than Deagol finding the Ring. I would have added Frodo's pity for Gollum, but I count that as part of 1.
Reviewing the list, it seems that I tend to favour courageous acts on the part of Hobbits.
But I think the consequnces of the acts and the effort involved are important to consider too.
PS - Thought of another one. Sam confronting and defeating Shelob. I would put it at number 6 or 7 on the list above. She was the spawn of Ungoliant, for goodness sake.