The Hourai Elixir (蓬莱の薬 Hourai no Kusuri) was created by Eirin Yagokoro using Kaguya Houraisan's power of the manipulation of eternity. Anyone who drinks this forbidden elixir will cease to age, never become ill, and will never be able to die. One will be as frail and vulnerable as before, but every injury shall cease; critical injuries will be healed in a matter of days. Complete destruction results in complete restoration.
The Hourai Elixir is permanent. According to Fujiwara no Mokou, the first taste of the medicine makes the drinker incapable of growing up or aging. The second taste makes them incapable of becoming ill. The third taste makes one completely immortal to the truest sense. The ingredients of the elixir are known only by Eirin Yagokoro, whose name points her out to be the "Mind of God", an omniscient being. Reportedly, the ingredients of the elixir remain in the liver of whomever drinks it. Eating the raw liver of an immortal is said to grant the same immortality as drinking the elixir, though it will not deprive one from their own because they have already been rendered deathless.
Because Eirin made the elixir the true elixir of immortality, she probably also made it irreversible, making its effects perfectly permanent. In her confrontation with Mokou, Alice Margatroid points out that the soul is made invincible. This, along with Yuyuko Saigyouji and Yukari Yakumo's scenario in Imperishable Night, makes it seem that the elixir itself isn't a power that keeps one immortal, but makes one an existence completely independent of even the concept of reality, thus making one not subject to the bounds of life, death, or even manipulation, although the pain of injuries can still be felt. As Alice has said, immortality casts away the physical body, and the soul is constantly reborn in new flesh. This medicine indeed makes one truly invincible, and forced to live out their existence forever. It's no wonder why this medicine is forbidden to the mysterious Lunarians, although it is unclear if this is also forbidden on the Earth.
In the past, this elixir has been used by Lunarians to test influential humans or to cause disturbances on Earth. The backstory of the Hourai Elixir is that Eirin made the medicine on Kaguya's request to drink it, resulting in her exile from the Moon. According to the first chapter of Cage in Lunatic Runagate, Eirin felt guilt over Kaguya's exile, and when the time came to bring Kaguya back to the Moon, Eirin deceived the Moon's emissaries and fled back to the Earth with Kaguya. With this tied to the traditional story of Kaguya-hime, Kaguya apparently left the elixir to the old bamboo cutter who raised her and the emperor whose heart she'd stolen before she left to the Moon, resulting in Mokou's backstory in the fourth chapter of Cage in Lunatic Runagate. There is no information as to where the elixir is presently or even if there is any left.
Since this elixir is forbidden on the Moon, and only Eirin is known to be capable of manufacturing it, and even then only with the help of Kaguya Houraisan's manipulation of eternity, it is understandably rare. Only four people are confirmed to have ever drunk it: