If a person did not complete that which was assigned to him in this world, his soul must return to the world in a different body until his rectification is complete. A person stops being reincarnated only once all parts of his soul have been rectified.

If a person’s nefesh [lowest level soul] was not entirely rectified the first time, and he passed away, that nefesh must return in a reincarnated form. This can occur several times, until all of its needs are entirely rectified. Then, even though it has been perfected, the ruaĥ [higher level soul] does not enter it unless there is a pressing need, due to the fact that the nefesh was rectified only through reincarnation…. Therefore, [the individual] needs to die, and his nefesh will then be reincarnated again and will receive its ruaĥ. Once the ruaĥ is rectified, he will [again] need to die and be reincarnated, and the neshama [highest level soul] will enter him, like the ruaĥ did. But if the ruaĥ was not rectified, the nefesh and the ruaĥ will need to be reincarnated together, perhaps several times, until the ruaĥ is rectified. The person will then die, and the nefesh and the ruaĥ will be reincarnated together with the neshama, [and this will be repeated] until all three are rectified. After this there will no longer be a need for reincarnation, because once the nefesh is rectified, the person is fully complete. (Rabbi Yitzĥak Luria, Sha’ar HaGilgulim, Introduction 2)

Further reading: Regarding into which body an individual whose soul has been reincarnated several times will be resurrected at the resurrection of the dead, see p. 298.

Every Jew bears a mark from God. For this reason, God does not give up on any Jew. He will reincarnate a person many times until that person rectifies himself and merits life in the World to Come.

You know the mystical teaching with regard to reincarnation, that one can be reincarnated from mineral to vegetable [or from vegetable to animal, or from animal to human], until he is finally rectified. This is as the Sage wrote in the mishna, “All of the Jewish people [have a share in the World to Come]” (Mishna Sanhedrin 10:1). Although one person rectifies himself within a short period while another takes a long time, they will [both] ultimately be counted among the righteous. For this reason, the Holy One, blessed be He, troubles Himself, as it were, with [reincarnating] wicked people, to rectify them. The Sage explains [in the continuation of the mishna], “As it is stated, ‘And your people, they are all righteous, they will inherit the land forever, the scion of My planting, My handiwork, to be adorned in splendor’ (Isaiah 60:21),” That is to say, after their sins are rectified [they will enter the World to Come]. Why? Because they are “the scion of My planting.” For He is eternal and they are the reflection of His essential light, and all who breathe, ultimately draw breath from God Himself. (Rabbi Yitzĥak Luria, Likutei HaShas, Avot)

The punishment of Gehenna serves to cleanse the soul of the impurities that have stuck to it, but it does not rectify the soul itself. For this, one needs to be reincarnated.

Gehenna has the power only to eliminate contamination [due to sinful acts]…but not to perfect [the soul]. Therefore, with regard to one whose sins did not allow him to perfect his soul with Torah and mitzvot, Gehenna cannot help him to do this, as it can be done only through Torah and mitzvot. Therefore, after [going through] Gehenna, where the contamination is weakened so that [in future] he will not be caught up in the web of [his] previous sins, he will need to be reincarnated in order to complete his soul…. Without Gehenna, it would be impossible for sinners to be reincarnated, as [the cleansing process of Gehenna serves to halt] the continuation of their sins. Gehenna completes the rectification only for someone who has already perfected his soul in this or a previous reincarnation, but who has become contaminated due to his sins. Gehenna cleans these out entirely, and he ascends, cleansed and in peace, to his place [in the Garden of Eden]. (Rabbi Shmuel Bornsztain, Shem MiShmuel, Exodus 5672)

Rabbi Elimelekh of Lizhensk was known for his great humility. He would often denounce himself harshly. After his death, the Maggid (preacher) of Koznitz said that when Rabbi Elimelekh came before the heavenly court, he was asked about his deeds in this world, and he cried out, “I did nothing! I did not pray! I did not study! Nothing, nothing, nothing!”

A righteous person loves life in this world only because it is a step from which to ascend to the World to Come. (Rav Se’adya Gaon, Emunot VeDe’ot 10)

The World to Come is immense and unending…a great light that does not resemble the light of this world. (Sefer HaYashar 14)

Further reading: By saying Kaddish, those who are still alive can benefit the soul of the deceased in the World to Come; see above, p. 29; A Concise Guide to Halakha, p. 504.