Archaic Chinese, which has been reconstructed and phonetically differs somewhat from any of the modern Sino-Tibetan languages.
Classical Mayan, which is slightly different from any of the modern Mayan languages.
I was under the impression both of these differed quite a lot from their descendants, since several centuries separate them.
I like very ancient languages
I believe that everyone should learn
What for? I'm not interested in Classics, so that rules out Latin and Ancient Greek. I'm not religious, so there go Hebrew, Koine Greek, Sanskrit and Pali. And most other dead languages are of limited use to non-specialists and lacking in resources.