To make the greatest possible effort.
do one's darnedest
To do one's utmost; to make every effort or to try every possible appr …
do one's damnedest
make every effort
To do one's utmost
give it one's best shot
To make one's best effort or attempt; to try as hard as possible.
go all out
To reserve nothing; to put forth all possible effort or resources.
give one's all
To lose one's life while making the utmost effort with full commitment …