I agree; it's probably not too often that something is truly unstattable. If you can pin down a definition of its capabilities, you can probably come up with stats. So it's probably best to talk not about whether it can be statted, but whether it should be.
In the case of the One Ring, you might say that the very best reason to not stat it (ie, keep it as a 'plot device') is that that is exactly what Tolkien did.
In the Hobbit, it was nothing more than a magical ring that made you invisible.
Then as the FotR opens, it is revealed that it is a ring of unspeakable power, one that is capable of ... what? Well, Tolkien was none too specific, but it granted control over the other Rings of Power. As the story unfolds, we learn that the One Ring has an inherent intelligence of sorts, and can sway the will of the bearer and others nearby. And it doesn't turn you invisible, it moves you into a sort of astral space which works like invisibility as far as most are concerned, but not with respect to Ringwraiths. And the Ring can make the bearer more powerful in other ways, too, if worn by someone who can tap that power.
I think there's a great reason Tolkien didn't insert a stat block for the One Ring into the Hobbit (I'm not counting 'third-party' RPGs here, which make great references but aren't 100% LotR). By rationing information, the plotline unfurled in a more interesting way. (Or maybe he hadn't fully planned out the LotR series yet, but we'll ignore that for now.)
If you give an item stats, then it becomes nothing more than a tool. Possibly a powerful tool, but a tool nonetheless. If you make it a 'plot device', then it becomes, in effect, an inanimate NPC ... which is exactly how something like the One Ring should be treated, in my opinion. It's still a tool, but now its a tool used to drive the plot, rather than mere gear.
(But by all means, go right ahead and write up stats. Then invoke its Plot Device aspect to add or change it as needed by the plot, as often as necessary!)