This is ATU 2044. Pulling up the Turnip. You can find out more at Wikipedia.
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THE BIG TURNIP - A RUSSIAN FOLKTALE
Once upon a time grandfather planted a turnip. The turnip grew and grew until it grew very big. By and by it was ready to eat.
So grandfather went out into the garden to pull up the turnip. He pulled and he pulled but he could not pull it up.
Then grandfather called grandmother to come and help him. Grandmother came running out and caught hold of grandfather. They pulled and they pulled but they could not pull up the turnip.
So they called to their granddaughter to come out and help them. The granddaughter ran out and caught hold of grandmother, grandmother clung to grandfather. They pulled and they pulled but they could not pull up the turnip.
Then granddaughter called the dog, "Little black dog, come and help us." The little black dog ran out and caught hold of granddaughter, granddaughter caught hold of grandmother, grandmother caught hold of grandfather. They pulled and they pulled but they could not pull up the turnip.
So the little black dog called the cat, "Little white cat, come and help us." The little white cat came running and caught hold of the little black dog, the little black dog caught hold of granddaughter, granddaughter caught hold of grandmother, grandmother caught hold of grandfather. They pulled and they pulled but they could not pull up the turnip.
Then the little white cat called the mouse, "Little gray mouse, come and help us." The little gray mouse came running and caught hold of the little white cat, the little white cat caught hold of the little black dog, the little black dog caught hold of granddaughter, granddaughter caught hold of grandmother, grandmother caught hold of grandfather.
They pulled and they pulled and they pulled up the turnip!
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Here is the Russian version as published by Afanasjev (online); it features a strange element in the pattern - an unknown word, нога. In his translation, Jack Haney speculates that it might be some kind of insect, so the final chain in this version is: grandfather, grandmother, granddaughter, dog, first bug, second bug, third bug, fourth bug, and finally the fifth bug, with whose help they finally pull the turnip out:
Посеял дедка репку; пошел репку рвать, захватился за репку: тянет-потянет, вытянуть не может! Созвал дедка бабку; бабка за дедку, дедка за репку, тянут-потянут, вытянуть не можут! Пришла внучка; внучка за бабку, бабка за дедку, дедка за репку, тянут-потянут, вытянуть не можут! Пришла сучка; сучка за внучку, внучка за бабку, бабка за дедку, дедка за репку, тянут-потянут, вытянуть не можут! Пришла нога (?). Нога за сучку, сучка за внучку, внучка за бабку, бабка за дедку, дедка за репку, тянут-потянут, вытянуть не можут! Пришла друга нога; друга нога за ногу, нога за сучку, сучка за внучку, внучка за бабку, бабка за дедку, дедка за репку, тянут-потянут, вытянуть не можут! (и так далее до пятой ноги). Пришла пята нога. Пять ног за четыре, четыре ноги за три, три ноги за две, две ноги за ногу, нога за сучку, сучка за внучку, внучка за бабку, бабка за дедку, дедка за репку, тянут-потянут: вытянули репку!
