Basic compounds
These are actually all a connector plus an auxiliary verb.
Joining with 고
Fixed pattern: Progressive tense (고 있다)
Fixed pattern: Want to (고 싶다)
Joining with 아/어
Compound words
Fixed pattern: as a favor (아/어주다)
Fixed pattern: try to (아/어 보다)
Fixed pattern: Should/must (아/어야 하다)