Catarrh is a Greek word wearing all its Greek markers at once. It has the rh that transliterates the Greek letter rho with its rough breathing, and it has it doubled at the end, which is why the spelling looks so unlikely for a word pronounced "ka-TAR." The Greek is katarrhous, literally "a flowing down," from kata, "down," plus rhein, "to flow" — an unusually vivid description of a head cold.
English took it in the late 1300s through Late and Medieval Latin, which is the standard route for Greek medical vocabulary: almost nothing came directly from Greek into English before the 1500s. The same rhein turns up in hemorrhage, diarrhea and rheumatism, and the same rh- opening in rhythm, rhetoric and rhinoceros. Once you know rh signals Greek, a whole quarter of the medical dictionary sorts itself out.
catarrh (n.) "disease characterized by inflammation of, and discharge from, a mucous membrane; a cold in the head or chest," late 14c., from Medieval Latin catarrus , from Late Latin catarrhus , from Greek katarrhous "a catarrh, a head cold," literally "a flowing down," earlier kata rrhoos , ultimately from kata "down" (see cata- ) + rhein "to flow" (from PIE root *sreu- "to flow"). Related: Catarrhal; catarrhous . also from late 14c.
catarrh came in through Greek, and its spelling follows that family's conventions.
What gives it away on the page
| rh-, -rrh- | Transliterates ρ with its rough breathing.rhythm, rhetoric, hemorrhage, catarrh |