Draupnir is a machine for producing arm-rings, and understanding why that is the most valuable object in Ásgarðr requires understanding how a Norse king paid people.
Skáldskaparmál states the function: every ninth night, eight rings of the same weight drip from it. The name is Draupnir, "the dripper," from a verb meaning to drip or drop.
Why a ring-machine is a weapon
Early Scandinavian rulers did not pay salaries. A lord kept a following by giving gold and silver away — arm-rings, neck-rings, and cut hack-silver by weight — and the entire vocabulary of Old Norse and Old English poetry for a good ruler reflects it. A king is a baugbroti, ring-breaker; a hringdrifi, ring-scatterer; in Old English, a beaga brytta and a sinces brytta, dispenser of treasure. The hall scene in Beowulf is one long description of distribution.
The arm-ring was the unit. Silver arm-rings of standardised weights survive in large numbers from Viking-age hoards, frequently cut into segments — because they were currency as much as ornament, and you paid by cutting off what you owed.
So a ring that produces eight rings of equal weight every nine nights is not a curiosity. It is an inexhaustible payroll. Odin's ability to maintain a following is, in the most literal terms available to the culture, self-financing.
Where it came from and where it went
It is one of the six treasures produced in the dwarf-wager that follows Loki's assault on Sif's hair — made by Brokkr and Eitri, in the same run of work that produced Mjölnir, and one of the three items the gods judged against those of the sons of Ívaldi.
Then Odin gives it away, at the worst possible moment. At Baldr's funeral, Snorri says, Odin laid on the pyre the gold ring called Draupnir, and afterwards it had the property that every ninth night eight rings of the same weight dripped from it.
Read carefully, that phrasing is ambiguous — Snorri introduces the multiplication property immediately after describing the ring going onto the pyre, which has led some readers to wonder whether the property was acquired in the fire rather than at the forge. He states the same property in Skáldskaparmál without the funeral context, so it is probably just clumsy sequencing.
What is not ambiguous is the gift itself. The chief god puts his source of wealth on his son's pyre, and gets it back only because Hermóðr rides to Hel and Baldr sends it up again as a keepsake — accompanied by Nanna's presents for Frigg and Fulla.
That is the whole story of the object: forged in a bet, given to a corpse, returned by the dead.
Its other appearance
Skírnismál has Skírnir offer Draupnir to Gerðr as the second inducement in the coercive courtship, describing it as the ring that was burned with Odin's young son. She refuses it, saying her father has gold enough for her to share with him.
That refusal is one of the great moments in Eddic poetry. The single most productive object in the mythology — the thing that generates the wealth kings are made of — is offered to a giant's daughter, and she says she does not need it. No one else in the corpus turns down an infinite money supply.
Snorri also names Draupnir as one of the things Freyr had, in some tellings, which is a genuine inconsistency: the ring is Odin's, offered on his behalf.
Rings and oaths
Beyond wealth, rings carry a legal charge in Norse practice. Oath-rings kept at sanctuaries and reddened with sacrificial blood are described in Eyrbyggja saga and the Landnámabók, and the Anglo-Saxon Chronicle records a Danish army in 876 swearing on a holy ring — a thing they had never done for any other people. Atlakviða has an oath sworn on "the ring of Ullr."
So a divine ring is not simply money. It sits at the intersection of the two things that held Norse society together: payment and sworn word. That Odin's ring reproduces itself, while an oath-ring binds, may be the same idea approached from two sides.
The ring that is not Tolkien's
Draupnir has had a modest but persistent afterlife: it appears in Marvel comics, in God of War Ragnarök as a ring with a multiplying function, and in a good deal of fantasy where the self-multiplying treasure is a standard motif. Tolkien's One Ring owes nothing to Draupnir and a great deal to Andvaranaut, the cursed ring of the Sigurd cycle — a distinction worth keeping, since the two Norse rings are frequently confused.
For its owner, see Odin; for the wager that produced it, Sif and Mjölnir; and for the Norse overview.
Do the rings Draupnir produces also drip?
Nothing in the sources says so, and the question is more interesting than it looks. If the copies inherited the property the growth would be exponential and the economy of Ásgarðr would collapse under its own gold within a year or two; if they do not, the ring is a steady annuity of eight units per nine nights. Snorri specifies only that the drops are "of the same weight," which is a statement about mass rather than about function. Medieval audiences appear not to have asked, and the specification of equal weight suggests the point was reliability of value, not compounding.
Why nine nights?
Nine is the pervasive ritual interval in Norse sources — nine nights on the tree, nine nights before the meeting at Barri, nine worlds, nine mothers, and the great sacrifices at Uppsala and Lejre held every ninth year with nine of each kind of victim, according to Adam of Bremen and Thietmar of Merseburg. Whatever the number meant, it structured time in a way the mythology applies consistently, and a ring that produces on that cycle is keeping to the calendar of sacrifice rather than to any natural period.