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I understand that 「の」 can be used to indicate possession or to change a noun to a modifier for another noun. But neither of those seems to be the case here.

What usage of 「の」 is being used here?

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終わりのない is not a complete sentence. This phrase can be used as a relative clause which modifies another noun.

  • 終わりのない話 a story which has no end; a never-ending story
  • 終わりがない話 a story which has no end; a never-ending story

As you can see, this type of の is a subject marker (just like が) which can be used only in relative clauses.

See: How does the の work in 「日本人の知らない日本語」?

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