give it a shot 是“试一试”，那 give it one's best shot 就是“尽最大的努力”

【例句】
We all gave our best shot to pass the exam.
我们都尽了最大的努力来通过考试。

I probably won't win the game, but I'll give it my best shot.
我可能赢不了这场比赛，那我会尽最大努力。