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.
Let's enjoy today to the fullest.
He shouted at the top of his lungs.
We used up the budget to the limit.
一杯 means 'full' or 'a lot', but 目一杯 intensifies it to 'to the absolute limit'.
Standard kanji-kana mixed spelling.
Variant spelling with いっぱい in kana; less common but still seen.
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'.