We found one dictionary that defines the word an egg's age:

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1. an egg's age: Wiktionary

Definitions from Wiktionary (an egg's age) ▸ noun:  (US, slang, dated) A very long time. ▸ Words similar to an egg's age ▸ Usage examples for an egg's age ▸ Idioms related to an egg's age ▸ Wikipedia articles (New!) ▸ Words that often appear near an egg's age ▸ Rhymes of an egg's age ▸ Invented words related to an egg's age