Emphatic adverb used to intensify verbs and adjectives. Common in both spoken and written Japanese. Often translates to 'very much', 'greatly', or 'to one's heart's content' depending on context.
Yesterday's party was a lot of fun.
この結果には大いに満足しています。
I am very satisfied with this result.
休暇中は大いに楽しんでください。
Please enjoy yourself to your heart's content during the vacation.
I enjoyed the holiday to the fullest.
I have high hopes for his future success.
I'm deeply grateful to the friend who helped me.
I agonized a great deal over my career path.