Gandalf says that Sauron put “a great part of his own former power” into the One Ring when it was forged:
“So it is now [said Gandalf]: the Nine he has gathered to himself; the Seven also, or else they are destroyed. The Three are hidden still. But that no longer troubles him. He only needs the One; for he made that Ring himself, it is his, and he let a great part of his own former power pass into it, so that he could rule all the other.”
J. R. R. Tolkien (1954). The Fellowship of the Ring, chapter 2. London: George Allen & Unwin.
The implication here is that beings like Sauron have a limited stock of power which gets spent in the making of magical artefacts like the One Ring, and this power is never replenished, so that these beings can only weaken and diminish with the passing of time: diminishing quickly if they spend that power or slowly if they conserve it. There is similar wording in The Silmarillion:
Now the Elves made many rings; but secretly Sauron made 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. And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow.
J. R. R. Tolkien (1977). ‘Of the Rings of Power and the Third Age’. In The Silmarillion. London: George Allen & Unwin.
This idea of beings spending a limited stock of power is ubiquitous in The Silmarillion. Míriel, a woman of the Noldor, spends her strength in the generation of her son Fëanor:
But in the bearing of her son Míriel was consumed in spirit and body; and after his birth she yearned for release from the labours of living. And when she had named him, she said to Finwë: “Never again shall I bear child; for strength that would have nourished the life of many has gone forth into Fëanor.”
Tolkien (1977), chapter 6.
When Melkor wishes to destroy the Two Trees of Valinor, he gives some of his power to the monstrous spider Ungoliant, and this permanently diminishes him:
In his right hand Morgoth held close the Silmarils, and though they were locked in a crystal casket, they had begun to burn him, and his hand was clenched in pain; but he would not open it “Nay!” he said [to Ungoliant]. “Thou hast had thy due. For with my power that I put into thee thy work was accomplished. I need thee no more. These things thou shalt not have, nor see. I name them unto myself for ever.”
But Ungoliant had grown great, and he less by the power that had gone out of him; and she rose against him, and her cloud closed about him, and she enmeshed him in a web of clinging thongs to strangle him.
Tolkien (1977), chapter 9.
Tolkien implies that Melkor was unable again to change the physical form he assumed during this episode:
Now Melkor came to Avathar and sought [Ungoliant] out; and he put on again the form that he had worn as the tyrant of Utumno: a dark Lord, tall and terrible. In that form he remained ever after.
Tolkien (1977), chapter 8.
After the destruction of the Two Trees, Yavanna says that she no longer has the power to remake them:
Yavanna spoke before the Valar, saying: “The Light of the Trees has passed away, and lives now only in the Silmarils of Fëanor. Foresighted was he! Even for those who are mightiest under Ilúvatar there is some work that they may accomplish once, and once only. The Light of the Trees I brought into being, and within Eä I can do so never again. Yet had I but a little of that light I could recall life to the Trees, ere their roots decay; and then our hurt should be healed, and the malice of Melkor be confounded.”
Tolkien (1977), chapter 9.
And Fëanor says the same about the Silmarils:
But Fëanor spoke then, and cried bitterly: “For the less even as for the greater there is some deed that he may accomplish but once only; and in that deed his heart shall rest. It may be that I can unlock my jewels, but never again shall I make their like; and if I must break them, I shall break my heart, and I shall be slain; first of all the Eldar in Aman.”
Tolkien (1977), chapter 9.
It is implied, though not stated directly, that the Valar no longer have the power that they had in the beginning of the world, when “Aulë at the prayer of Yavanna wrought two mighty lamps for the lighting of the Middle-earth” and “Varda filled the lamps and Manwë hallowed them, and the Valar set them upon high pillars” (chapter 1). The Trees are a minor accomplishment compared to the Lamps, and the Sun and Moon mere remnants of the Trees.
Like Melkor after the darkening of Valinor, Sauron can no longer assume a fair shape after the fall of Númenor:
[In Mordor] now he brooded in the dark, until he had wrought for himself a new shape; and it was terrible, for his fair semblance had departed for ever when he was cast into the abyss at the drowning of Númenor.
Tolkien (1977). ‘Of the Rings of Power and the Third Age’.
Tolkien says in a letter that
Sauron was, of course, ‘confounded’ by the disaster, and diminished (having expended enormous energy in the corruption of Númenor).
J. R. R. Tolkien (14 October 1958). Letter to Rhona Beare. Number 211 in Humphrey Carpenter, ed. (1981). The Letters of J. R. R. Tolkien. London: George Allen & Unwin.
So reading all this back into The Lord of the Rings, we can guess that Sauron put so much of his power into the One Ring, that even if he wanted to, he could no longer make another, just as the Valar could not remake the Lamps, nor Yavanna the Trees, nor Fëanor the Silmarils.
This theme of diminishment, of falling away from an original perfect state, is Tolkien’s main theme in The Silmarillion:
The main body of the tale, the Silmarillion proper, is about the fall of the most gifted kindred of the Elves, their exile from Valinor (a kind of Paradise, the home of the Gods) in the furthest West, their re-entry into Middle-earth, the land of their birth but long under the rule of the Enemy, and their strife with him, the power of Evil still visibly incarnate. It receives its name because the events are all threaded upon the fate and significance of the Silmarilli (‘radiance of pure light’) or Primeval Jewels. By the making of gems the sub-creative function of the Elves is chiefly symbolized, but the Silmarilli were more than just beautiful things as such. There was Light. There was the Light of Valinor made visible in the Two Trees of Silver and Gold. These were slain by the Enemy out of malice, and Valinor was darkened, though from them, ere they died utterly, were derived the lights of Sun and Moon. (A marked difference here between these legends and most others is that the Sun is not a divine symbol, but a second-best thing, and the ‘light of the Sun’ (the world under the sun) become terms for a fallen world, and a dislocated imperfect vision).
J. R. R. Tolkien (1951). Letter to Milton Waldman. Number 131 in Carpenter (1981).
The idea of a falling away from an original paradise is commonplace in mythology. There is the story of Eden in the book of Genesis, and the myth of the “golden age”, for example as told by Hesiod:
First of all the deathless gods who dwell on Olympus made a golden race of mortal men who lived in the time of Cronos when he was reigning in heaven. And they lived like gods without sorrow of heart, remote and free from toil and grief: miserable age rested not on them; but with legs and arms never failing they made merry with feasting beyond the reach of all evils. When they died, it was as though they were overcome with sleep, and they had all good things; for the fruitful earth unforced bare them fruit abundantly and without stint. They dwelt in ease and peace upon their lands with many good things, rich in flocks and loved by the blessed gods.
Hesiod. Works and Days, lines 109–120. Translated by Hugh G. Evelyn-White. gutenberg.org
Hesiod goes on to describe how the golden age was followed by a silver age, a bronze age, an age of heroes, and finally an iron age, in which “men never rest from labour and sorrow by day, and from perishing by night”.