Я чуть не забыл зарядку дома, но вернулся за зарядкой.

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Breakdown of Я чуть не забыл зарядку дома, но вернулся за зарядкой.

• дома: at home
• но: but
• за: for

Questions & Answers about Я чуть не забыл зарядку дома, но вернулся за зарядкой.

What does чуть не mean here, and why is there не if the meaning is positive?

Чуть не + past tense is a fixed pattern meaning almost (something happened, but in the end it didn’t).
So Я чуть не забыл = I almost forgot.
The не is part of the construction: grammatically it looks like negation (didn’t forget), but the whole phrase means came close to forgetting.

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Why is it забыл (perfective) and not забывал (imperfective)?

Забыть / забыл (perfective) is used for a single completed event (or a near-event with чуть не): one moment where you nearly forgot.
Забывать / забывал (imperfective) would suggest a habit/repeated situation or a process (e.g., I used to forget, I was forgetting), which doesn’t fit as well here.

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What does дома mean grammatically? Is it a case?

Дома is an adverb meaning at home. Historically it relates to the genitive form, but in modern Russian you can treat дома as a set adverb (like внутри, снаружи).
So забыл зарядку дома = forgot the charger at home.

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