Ecstasy seems like it should have some roots in it. Perhaps stas- as in stasis? They have very different meanings though, so how would you get from stasis to ecstasy?
Dictionary.com confirms the guess of stas- as a root. Apparently ecstasy comes from the Greek ékstasis meaning "displacement" or "trance." The Greek ékstasis comes from the prefix ek- and the root stasis, which has its modern meaning. Ek- is a form of the familiar prefix ex- meaning "out" or "from." Thus ecstasy means "out of stasis," i.e., "out of balance/equilibrium (emotionally)," which is what it means today, though I think it has acquired a positive connotation that it did not always have.
The ec- form of ex- still appears in modern English; for example, eccentric, which presumably means, literally, "out of center."