The riddles of Bilbo and Gollum
JRR Tolkien

Gollum:
What has roots as nobody sees,
Is taller than trees,
Up, up it goes,
And yet never grows?
[A mountain]

Bilbo guesses this one immediately. (It is asked before the game officially begins -- in fact, Gollum suggests the game because Bilbo says it's easy.)

Bilbo:
Thirty white horses on a red hill,
First they champ,
Then they stamp,
Then they stand still.
[Teeth]

This is an old one, Gollum is almost disgusted to be asked a 'chestnut', as he puts it.

Gollum:
It cannot be seen, cannot be felt,
Cannot be heard, cannot be smelt.
It lies behind stars and under hills,
And empty holes it fills.
It comes first and follows after,
Ends life, kills laughter.
[Dark]

Since the answer is 'all around him', Bilbo guesses it immediately.

Bilbo:
A box without hinges, key, or lid,
Yet golden treasure inside is hid.
[An egg]

Gollum takes a while, but finally remembers thieving from nests and teaching his grandmother to suck eggs.

Gollum:
Alive without breath,
As cold as death;
Never thirsty, ever drinking,
All in mail never clinking.
[Fish]

Bilbo almost gets eaten at this one, but luckily a fish splashes out of the water and on to his foot, giving him his clue.

Bilbo:
No-legs lay on one-leg,
two-legs sat near on three-legs,
four-legs got some.
[Fish on a table,
man on a stool,
cat gets the bones]

Since the earlier one was 'fish', the no-legs part was easy, and the rest followed.

Gollum:
This thing all things devours:
Birds, beasts, trees, flowers;
Gnaws iron, bites steel;
Grinds hard stones to meal;
Slays king, ruins town,
And beats high mountain down.
[Time]

Bilbo thinks of giants, and cannot find one who has done all this -- wants to ask for more 'time', can't get all the words out and simply screams 'time', which saves him.

The final victory isn't, ironically, with a rhyme at all, but 'What have I got in my pocket?' (He is talking to himself and Gollum thinks it's a riddle, can't guess, and Bilbo wins the game. Gollum doesn't show him the way out, though, Bilbo follows him secretly.)