In later, more localized adaptations of Buddhism, there are many hells, or Narakas, where those who racked up particularly bad karma were reborn, where they live, suffer, die, and are reborn again for many kalpas (eons) on end until they have worked off all their bad karma. But the lowest hell, Avici, is reserved for those who commit one or more of the Five Grave Offenses, the personal Moral Event Horizons of the religion: intentionally murdering one's father, intentionally murdering one's mother, killing an arhat (enlightened being), shedding the blood of a Buddha, and causing a schism in the sangha (the community of Buddhist monks and nuns). Existence in the Avici hell lasts the longest out of all of the other hells put together, such that it is often known as "the non-stop way."
Avici hell means "without waves". One can translate it to The Ceaseless. With a cosmology where a trillion of trillion years is a mere metric for time, this should clarify that the cosmic judgment of karma only reserves Avici for Complete Monsters only.
Also, considering the cyclic nature of Buddhist cosmology, even sins fit for Avici is not truly irredeemable. Still, that is only theoretical. Nobody who has fallen into Avici, since the beginning of reality an infinity years ago, has been redeemed yet.
However, there is dispute about when this idea originated. Traditionally, it comes from the story of Devadatta, a monk who killed his father, twice assaulted the Buddha, and split the sangha. But there are at least two versions of the story of Devadatta - one of which has him being consigned to a very long stay in Avici and one of which has him repenting and achieving some level of enlightenment. Some historians date the story of Devadatta to a hundred years or more after the Buddha died, which would make it a later addition.
In older Catholic teaching and Word of Dante, the only truly unforgivable sin is to commit self-murder (suicide), since you're not alive anymore to be forgiven afterward. In recent years (since the Second Vatican Council), the Roman Catholic Church has ceased to teach that suicide is unforgivable. Quoting from the Catechism of the Catholic Church, sec. 2283: "We should not despair of the eternal salvation of persons who have taken their own lives."
The Catholic teaching on mortal sin is that it must be a deliberate act with the knowledge that it is a mortal sin. A completed suicide is unforgivable for someone who understood it to be a mortal sin and freely chose to commit it anyway. Not so for someone who is clinically insane and therefore lacking full control over his/her faculties. Some Christians hold that nobody who actually wants to die could be considered sane in any normal sense, so suicide is never actually a sin.
Further, this analysis assumes that the death is instantaneous. A mortally wounded person is still capable of a deathbed conversion, even if the mortal wound was self-inflicted. Confession and/or the Anointing of the Sick can be validly received to effect the forgiveness of mortal sins provided that the penitent has (at a minimum) sorrow for his/her sins arising out of fear of God's wrath.
In the book of Revelation, taking the "mark of the beast" and worshiping his image is considered a Moral Event Horizon.
According to various verses in the New Testament, rejecting the offer to accept Jesus as savior and rejecting His gift of forgiveness and salvation can be seen as a Moral Event Horizon. Note that this happens during the person's death.
In Greek Mythology, you can do almost anything-including murder and rape-and still be considered a hero, but there are three exceptions, and any hero who committed one of these three sins would lose their title: Hubris, impiety, and violation of xenia.
Judaism has some sins which can only be completely atoned for through death - "desecration of the Divine Name", which means hurting God's reputation, being one of them. One interpretation is that it is considered so heinous because it can cause others to turn away from God and to sin.
It should be noted that this is not really an example of this trope, precisely because death does atone for the sin, meaning that the person who committed it is not eternally damned. According to Judaism, the MEH does not exist, because God's mercy is infinite: "The Gates of Repentance are always open," as Rabbi Shmuel bar Nahman put it. It should also be noted that death completes the redemption; one does have to actually repent, which begins the sinner's atonement, but in the case of truly horrible sins, it is death that completes it.
It's hard to pinpoint exactly when King Saul crossed it, but there are two candidates.
Until the second time David failed to show up after being invited to one of King Saul's banquets, Jonathan, who Saul was certain was first in line to succeed him, thought that King Saul was simply out of his mind. But when Jonathan gave him a cover story that David wanted to spend some time with his family, King Saul snapped, called him a son of a bitch, and–more heinous than that–actually tried to kill his own son.
If that doesn't sell you on the point that King Saul had sealed his own fate, his unrelenting destruction of a city of priests and the execution of most of the same, including their families–yes, women and children included–certainly will. How unnecessarily cruel was it? Only one of his men was willing to carry out the grisly mass murder.
Similarly the sons of perditionnote Those who either, uring their pre-mortal life, aligned themselves with Satan's plan of salvation, or, in mortal life, have "denied the Holy Ghost" i.e. rejecting and denying Christ after receiving a personal witness and a "perfect knowledge" of Jesus (see "blasphemy against the Holy Ghost" above) are not to benefit either of the celestial, terrestrial or telestial glory but instead are cast into the outer darkness with Satan and his angels.