the hundredth year
Literal meaning: the 100th year of an era, period, or age. Often used in historical or chronological contexts.
This town reached its hundredth year.
end of one's rope; out of luck; rare chance; unusually good fortune
Idiomatic use derived from the idea of a once-in-a-century event. Sense 2 (out of luck) is often used in the phrase 百年目の不作 (a once-in-a-century bad harvest). Sense 3 (rare chance) is less common and may appear in expressions like 百年目の幸運 (once-in-a-century luck). Both are figurative and somewhat literary.
This year's poor harvest is said to be a once-in-a-century event.
Such good fortune is a once-in-a-century occurrence.