The boon Mahisha asked for (his birth, and the demon at her feet)
Rambha and Karambha pray for sons; Indra drowns Karambha as a crocodile; the Fire grants Rambha a son by a she-buffalo, and Mahisha is born from her on Rambha's pyre. Grown, he does tapas to Brahma, asks to die by no god, demon or man, and laughs at the thought of a woman. He conquers earth, defies Indra, and in the end the Goddess takes his head. The Kalika Purana adds his three deaths and the place at her feet he asked for.
Vyasa tells Janamejaya how Mahisha came to be. Danu's sons Rambha and Karambha, childless, do tapas at Pancha Nada, Karambha in the water and Rambha under a yaksha-haunted peepul tree before a fire; Indra as a crocodile kills Karambha; Rambha would cut off his own head but the Fire stops him and grants a world-conquering son by any female he desires. Rambha couples with a she-buffalo, takes her to Patala, is gored to death by a bull buffalo; the yakshas kill the bull and burn Rambha, the she-buffalo enters the pyre, and Mahisha comes out of it, with Rambha reborn as Raktabija. Mahisha's ten thousand years on Sumeru win Brahma's boon: not immortality, which Brahma refuses as against the law of birth and death, but death from no god, demon or man, women being in his eyes too weak; Brahma says a woman will kill him. He subdues the earth, appoints Chikshura, Tamra, Asiloma, Bidala, Trinetra and Vaskala, and orders Indra to quit heaven, serve or fight; Indra calls him a grass-eating buffalo's son and threatens to make a bow of his horns; Mahisha, enraged, calls up the danavas. In 5.18 the Goddess kills his commanders, he changes shape (man, lion, elephant, sharabha, buffalo), she drinks wine and beheads him with her discus; in 5.19 the gods praise her and she vanishes after promising to come when called.
The Kalika Purana: three deaths and the place at her feet
Northeast India · asShakta🪔 Durga Puja
The Kalika Purana of Kamarupa says the Goddess killed Mahisha three times: as eighteen-armed Ugrachanda at the first creation, as sixteen-armed Bhadrakali in the second, and as Durga at last (digest citing Kalika Purana 60 and 61). Later tellers of the text add that between his deaths Mahisha dreamed of dying by a dark goddess, worshipped Bhadrakali, and when she appeared sixteen-armed and showed him his earlier deaths he asked two things: a share of the offerings made to her, and never to be parted from her feet while the sun lasts. She answered that the shares were already given to the gods, but that when she killed him she would hold him under her foot and the honour given to her feet would reach him. This is the reason given for the demon's place under her foot in every image, and for his going to the river with her.
What is different here:
Three killings across three creations instead of one; the demon's own devotion and his requested place at her feet, which the Devi Bhagavata does not have (it only says her slain enemies go to heaven).
Sources: Devi Bhagavata Purana, book 5, chapters 2 and 3 (Swami Vijnanananda, 1921; Wisdom Library); Devi Bhagavata Purana, book 5, chapters 18 and 19, the killing of Mahisha and the gods' praise (Vijnanananda; sacred-texts mirror); Purankosh: Mahishasura in the Kalika Purana (Ugrachanda, Bhadrakali, Durga), with verse citations; Hindu Blog: the story of Mahishasura's boon and his three births in the Kalika Purana (popular retelling); Mahishasura — Wikipedia