If you say that you will do your damnedest to achieve something, you mean that you will try as hard as you can to do it, even though you think that it will take a lot of effort.
例句
I did my damnedest to persuade her.我尽了最大的努力去说服她。
ADJ-SUPERL
令人万分惊奇的;极为古怪的
If you say that something is the damnedest thing, you are emphasizing that it is surprising or odd.
例句
Today I heard somebody say the damnedest thing about Cross...今天我听人说了关于克罗斯的最令人难以置信的话。You pick the damnedest places to eat!你可真会挑地方吃饭！