adverb, noun which may take the genitive case particle 'no', na-adjective, noun
to the limit; as much as possible; with all one's might
Expresses doing something to the fullest extent, using maximum effort, capacity, or volume. Often used adverbially with に or as a の-adjective.
今日は目一杯楽しもう。
Let's enjoy today to the fullest.
彼は目一杯の声で叫んだ。
He shouted at the top of his lungs.
予算は目一杯使ってしまった。
We used up the budget to the limit.
Compound of 目 (め, 'eye' or 'limit') and 一杯 (いっぱい, 'full'). The 目 here likely indicates a boundary or limit, giving the sense of 'up to the brim' or 'to the maximum'.