"되게" may be a conjugation of 되다, or an adverb derived from it.
* 되다
1. [verb] become; turn into.
- 훌륭한 인물이 되었다: Became a great figure.
- 비참한 신세가 되었다: Fell into a miserable condition.
2. [verb] be good enough; suffice; fit/conform the requirement.
- 됐어요: That's good; That's enough. (돼 = 되어; 됐어요 = 되었어요)
- 이렇게 하면 돼: Do this to make it work.
- 나는 가야 돼: I have to go.
3. [adjective] thick (a mixture, dough, etc); tough, difficult (a person, situation).
- 반죽이 너무 되다: The mixture is too thick.
- 그 녀석 주먹이 되다: His got a stinging punch.
* 되게
1. a conjugation of 되다:
- 안 되면 되게 하라: Take what fails and make it work.
- 사람 낚는 어부가 되게 하리라: I will make you become fishermen fishing for people.
2. adverb for "very much", "exceedingly":
- 사람이 되게 좋다: (He/she) is so very nice.
- 되게 잘난 척하네: (He/she) makes such a big deal of him/herself.
- 길을 잘못 들었다가 되게 고생했다: Had a real hard time after taking the wrong route.