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…