Something that's exasperating is frustrating or irritating. Your exasperating roommate might talk loudly on the phone late into the night and then wake up early to make his breakfast and leave your sink full of dirty dishes.
Traffic jams can be exasperating, especially if you're late for an important meeting, and chattering airplane seat mates are especially exasperating on very long flights. Things that make you annoyed are exasperating, and the word stems from the Latin exasperare, "irritate, provoke, or make rough," from the roots ex-, "thoroughly," and asper, "rough."