Exasperation is the worn-out irritation of dealing with something impossible. Past annoyance, and not quite anger.
The exhaustion is what defines it, since exasperation arrives after repeated failure rather than after one offence. It appears in sighing with exasperation and in exasperation as a phrase meaning because there was nothing else left. The verb exasperate is what the situation does to you. Frustration is the near synonym and is about being blocked rather than about being worn down.
1nounexasperationWeary irritation after repeated failure or annoyance.
“He threw the manual down in sheer exasperation.”
Also accepted in games: frustration