By all means; definitely — an emphatic form of ぜひ used to strongly urge or invite someone.
パーティーにぜひぜひ来てください！
Please do come to the party by all means!
このレストラン、ぜひぜひ試してみてください。
You should definitely try this restaurant.
興味があればぜひぜひ連絡してください。
If you're interested, please do contact me.