An adverb meaning 'extremely' or 'very', used to emphasize degree. (Note: the same-sounding 'gokugoku' can also be an onomatopoeia for gulping.)
彼の説明はごくごく簡単だったので、誰でもわかった。
His explanation was extremely simple, so anyone could understand it.
これはごくごく基本的なルールだから、しっかり覚えておいてください。
This is a very basic rule, so please remember it well.
その症状はごくごく軽い風邪のようだ。
That symptom seems very much like a mild cold.