As for as I know, the concept of exact seven re-incarnations are not described in any scripture (though I have not read all scriptures). But seven births is used as a standard notation.
I am giving below some examples from where seven births concept may be taken.
- Mahabharata, Santi Parva
Kundrika and king Brahmadatta of great energy, 2 repeatedly thinking of the sorrow that attends birth and death, attained to that prosperity which is acquired by persons devoted to Yoga, in course of seven births, in consequence of my favour.
- Devi Bhagavatam, Ninth Book, Chapter XXIV
So if one does not offer the Tulasî leaves in the conchshell, for seven births he remains without his wife and he becomes diseased.
- Devi Bhagavatam, Ninth Book, Chapter XXXIV
Dharma Râja Yama said :-- O Fair One! If, in this Bhârata, any murderer, merciless and fierce, kills any man, out of greed for money, he goes and miserably dwells in the Asipattra hell for fourteen Indra’s life periods. And if that murderer kills a Brâhmana, he lives in that hell for one hundred manvantaras. While in hell, his body becomes fiercely cut and wounded by the swords. There My messengers chastise him and beat him and he cries aloud and passes his time without any food. Then he becomes born for one hundred years as a Manthâna insect, for hundred births as a boar, for seven births as a cock; for seven births as a fox, for seven births as a tiger; for three births, as a wolf; for seven births, as a frog; then as a buffalo when he becomes freed of his sins of murders. If anybody sets fire to a city or a village, he will have to live in Ksuradhâra Kunda for three yugas with his body severed. Then he becomes a Preta (disembodied spirit) and travels over the whole earth, being burnt up with fire. For seven births he eats unclean and unholy food and spends his time as a pigeon. Then for seven births he becomes diseased with a severe colic pain, for seven births as a leper; when ultimately he gets a pure human body. If anybody whispers in one’s ear another’s calumny and thus glorifies himself and abuses and vilifies the Devas and Brâhmanas, he goes and remains in Sûchî Kunda for three Yugas, and he is pierced there by needles. Then he becomes a scorpion for seven births, a serpent for seven births, and an insect (Bhasma Kîta) for seven births; then he gets a diseased human body when, at last, he becomes purified. If anybody breaks into another’s house and steals away all the household articles, cows, goats or buffaloes, he goes to Gokâ Mukha Kunda where faeces are like cow’s hoofs, there, beaten by My servants, for three Yugas. (Gokâ is Goksura, hoof of a cow). Then, for seven births, he becomes a diseased cow; for three births, a sheep; for three births, a goat; and finally he becomes a man. But in this man-birth he is born first as diseased, poor, deprived of wife and friends, and a repenting person; when ultimately he is freed of his sin. If anybody steals any ordinary thing, he goes to Nakra Mukha Kunda and lives there for three years, greatly tormented by My messengers. Next for seven births, he becomes a diseased ox. Then he attains a very diseased man-birth, and ultimately he is freed of his sins. Such are the horrible results. If anybody kills a cow, elephant, horse, or cuts a tree, he goes to Gaja Dams’a Kunda for three yugas. There he is punished by My messengers freely by the teeth of elephants. Then he attains three elephants’ births, three horse-births; then he becomes born as a cow and ultimately he is born a Mlechcha when he becomes pure. If anybody obstructs any thirsty cow from drinking water, he goes to Krimi Kunda and Gomukha Kunda filled with hot water and lives there for one manvantara. Next when he attains a human birth he owns not any cattle nor any wealth; rather he is born as a man, very much diseased, in low castes, for seven births when he becomes freed.
So, it may be people have taken the concept of seven re-incarnations by mistake.