Several characters refer to their desire to use the power of the ring for good but know that they would be corrupted by its use.
What other powers can be drawn from the ring, or are they all referring simply to the ability to become invisible?
Several characters refer to their desire to use the power of the ring for good but know that they would be corrupted by its use.
What other powers can be drawn from the ring, or are they all referring simply to the ability to become invisible?
The notion of power in Tolkien is quite difficult to understand. In his conception, power is native to a person, and not something that is learned or even exercised. Witness Galadriel's confusion over Sam's use of the word "magic".
What seems to be the case is that all the rings mainly enhance the user's native power, rather than conferring new powers on them. The dwarven rings helped the owner to amass gold, and the elven rings helped the owner to preserve things. So it is likely that the effects of the One Ring depended mainly on the user.
Indeed, it might even be argued that the invisibility itself is an example of this: Hobbits had the native ability of stealthiness, and as Tolkien says in the introduction, to us this ability often seems "magical". So perhaps the real invisibility given by the Ring is just an enhancement of this ability. Note that the only non-Hobbit we see using the ring is not made invisible, although since this is Tom Bombadil, it might not be a good example since he has abilities of his own.
There are many.
Sam showed the ability to understand the Black Speech when he wore it - a language he had never spoken before. His auditory perception was greatly increased as well, but his vision was shifted into the unseen world.
The invisibility power is known.
The ability to penetrate disguises of other ring bearers. (cf. "The Mirror of Galadriel"). Also the ability of Frodo to perceive the true forms of the Nazgûl.
The ability to perceive the thoughts of (and if strong enough to control) other ring bearers. (ibid) (This was it's primary purpose and why it was sub-created.)
The ability to inspire fear in the servants of the Dark Lord. (cf. "The Tower of Cirith Ungol," "Mount Doom")
In general the Ring also greatly amplified the inherent powers of the wearer. (e.g. Sam's otherwise incredible ability to defeat Shelob single-handedly.)
The downsides seem to mainly be that it attracts servants of the dark lord, causes the wielder to have delusions of grandeur, slips off when least expected and inflames treachery and lust for power in others and the wielder.
In the story, of course, the Ring is in the hands of a relatively low powered wielder. In the hands of Saruman, Galadriel, Aragorn or Gandalf (let alone Sauron!) it would be far more terrible. (But not limitless; Sauron held the One when he was defeated in the War of the Last Alliance.)
The One Ring has plenty of powers other than invisibility.
It was originally intended to be able to rule and control the minor Rings, and could definitely be put to that use:
Sauron made the One Ring to rule all the others, and their power was bound up with it, to be subject wholly to it and to last only so long as it too should last. [...] while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them. (Silmarillion)
As WOPR mentions, it may give the power to speak and understand otherwise unintelligible languages:
Perhaps the Ring gave understanding of tongues, or simply understanding, especially of the servants of Sauron its maker, so that if he gave heed, he understood and translated the thought to himself. (Return of the King)
The Ring could be used to dominate lesser beings (Númenóreans in the case of Sauron):
He naturally had the One Ring, and so very soon dominated the minds and wills of most of the Númenóreans. (Letter 211)
It could also choose to "leave" or "abandon" its bearer and attempt to return to Sauron, as we see in a number of cases.
Isildur:
the Ring betrayed him and avenged its maker. (Silmarillion)
Gollum:
“A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. [...] It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him.” (Fellowship of the Ring)
Returning to Sauron:
Even from afar he had an effect upon it, to make it work for its return to himself. (Letter 246)
We also see plenty of examples of the Ring deceiving and corrupting its bearer and others, as well as references to its ability to do so - Gollum, Bilbo, Boromir, Frodo and even Sam all fall under its thrall to some extent, and Gandalf, Elrond and Galadriel all warn of it.
Finally, it may have all the powers of the lesser Rings, as Tolkien wrote of:
the Ruling Ring that contained the powers of all the others (Letter 131)
Longevity is another power. I'm not sure it's ever said how old Gollum was exactly, but he was way older that he should have been. And Bilbo... "Time wore on, but it seemed to have little effect on Mr Baggins".
The Ring's powers can be most clearly seen from the spell/poem used in its creation: Ruling, Finding, Summoning and Binding. We know that the dark tower was made using the binding power. Characters had ability to hear, see and understand things they couldn't before — the finding power.
Evil things were summoned to the ring-bearer's presence. The last thing you ever wanted to do was make a bargain with the ring wielder. Isildur unwittingly used the power of the Ring to curse a whole race of people undead for violating an oath and Gollum was killed by the Ring for breaking his oath to Frodo — one instance of the Ring's power working against itself.
All of the rings of power were originally intended for use by immortals - Elves and Maia, so their effects on mortals were a little unpredictable. The invisibility power was likely a manifestation of Frodo/Bilbo/Sam/Sméagol being overshadowed by the Ring's power and part of the Ring's finding power. The wearer could find things out about others but was secure from being found out himself.
Longevity is a side effect of the ring of power being originally intended for immortals - elves and maia. It certainly would not have been part of the ring's intended power. What need would an elf or a maia have of a ring to make him immortal?
Invisibility is only a side-effect of the Ring. The Ring takes the bearer into the spirit world - rendering the bearer invisible to those not in that world but rendering the bearer hyper-visible to entities of spirit alone, like Sauron and the Nazgûl. The more advanced powers of the Ring also involve the spirit world - being able to influence and dominate others, immortality, etc.
When Men wore the Ring, they would be partly "shifted" out of the physical realm into the wraith-world. A side effect (but usually the first noticed) of the Ring was that it made the wearer invisible to most inhabitants of the physical realm (but highly visible to the Nazgûl), it dimmed the wearer's sight, and it sharpened his hearing. When Men wore the ring, only their faint and shaky shadow could be seen, and only in the full light of the sun.[9] This "shadow world" was the world which Wraiths inhabited, but also where the Calaquendi (Elves of Light) lived at the same time as the normal world and held great power, as was evidenced by Frodo viewing Glorfindel at the Ford of Bruinen near Rivendell[12] and later explained by Gandalf.[13]