[Alcuin]

In Letter 211, Tolkien wrote,

Sauron was first defeated by a “miracle”: a direct action of God the Creator, changing the fashion of the world, when appealed to by Manwë… Though reduced to “a spirit of hatred borne on a dark wind”, I do not think one need boggle at this spirit carrying off the One Ring, upon which his power of dominating minds now largely depended.

You might ask why Sauron did not or could not do this after Elendil and Gil-galad killed him on the slopes of Orodruin, preventing Isildur from cutting off his finger and taking the One Ring.
In Letter 199, Tolkien wrote that the Ainur “often took the form and likeness” of Elves and Men, “especially after their appearance.”

…Sauron appeared in this shape. It is mythologically supposed that when this shape was “real”, that is a physical actuality in the physical world and not a vision transferred from mind to mind, it took some time to build up. It was then destructible like other physical organisms. But that … did not destroy the spirit, nor dismiss it from the world to which it was bound until the end. After the battle with Gil-galad and Elendil, Sauron took a long while to re-build, longer than he had done after the Downfall of Númenor (I suppose because each building-up used up some of the inherent energy of the spirit, which might be called the “will” or the effective link between the indestructible mind and being and the realization of its imagination). The impossibility of re-building after the destruction of the Ring, is sufficiently clear “mythologically” in the present book.

Originally Posted by Kuruharan

This begs the question of why did they make so many and who were the original intended recipients.

The smiths of Eregion made them for themselves.

The original population of Rivendell was made up of survivors of Sauron’s assault upon Eregion in the Second Age: Gil-galad sent an expeditionary force from Lindon under Elrond to bolster Eregion’s defenses until the Númenórean fleet arrived. Sauron’s armies swept into and to the west of Eregion, to block both any further assistance from Gil-galad and any retreat to him from Eregion. Elrond led the survivors to Imladris; the context as I recall it indicated that these were not very numerous.

Before the War between Sauron and the Elves, there were two sizable Noldorin polities. The greater and more numerous was Lindon, where there were also a large number of Sindar. The smaller was Eregion, where the Noldor established what we might call a “colony” outside the immediate control of Gil-galad. Depending upon which storyline you follow, its settlement might have been instigated by Galadriel, Celebrimbor, or both acting together. Though nominally under the suzerainty of Gil-galad as Noldorin High King, Eregion was essentially politically independent. After Gil-galad prevented Annatar (Sauron in disguise) from entering Lindon, Celebrimbor and the Mírdain (society of smiths) welcomed him in Eregion. When Galadriel, who was in Eregion, objected to the Mírdain’s welcome of Annatar, she was ostracized and left; she did not return to Lindon, but rather to Lothlórien, a kingdom of Silvan Elves ruled by Sindar.

Though it isn’t stated, it is very difficult to resist these conclusions

• The Noldor were divided politically from the beginning. The Sindar said the Noldor returned to Middle-earth for more room to argue.
• It sounds as if a lot of the surviving Fëanorians still in Middle-earth followed Celebrimbor and Galadriel to Eregion, where they exercised considerable political independence, including a close alliance with the Dwarves of Khazad-dûm.
• Though smaller in population than Lindon, the Elves of Eregion were numerous, certainly many times the number of refugees who founded Rivendell. It had a city, Ost-in-Edhil, well-executed roads, and seems to have had numerous other settlements as well. It sounds as if the Tower of the Mírdain was a sort of guild hall, with most of the shops and smithies of its members throughout the city and country.
• The appearance of Annatar further fractured Noldorin polity, with Galadriel splitting off and moving even further [west [sic]] east [hat tip Belegorn], over the Misty Mountains.

We aren’t given any indication how many Mírdain there were, but it seems they were rather numerous. Proud and self-willed, it isn’t difficult to see that they would all want Rings of various sorts, and of course these Rings should enable the Elves wielding them to shrug off or even forestall the effects of time, as well as enhance other skills or abilities the smiths found useful or desirable. Celebrimbor made the Three by himself; you may well ask for whom he made them, and why did he make three instead of only one? I cannot answer that, except that it permits Tolkien to tell a great story! You might also argue whether he alone made them, or other smiths assisted. But there is no reason for the proud Smiths of Eregion to craft rings for the use of Mortal Men or Dwarves, and there was no particular reason for them to stop making Great Rings until they bored of the effort. After all, they were Noldor, and it was Art. Useful Art, but Art. Why did they make so many? I don’t know, but I think you might as well as ask a famous painter why he continues to paint: He enjoys it. That’s what he does.