I'd personally choose to learn Japanese or Korean, because it's the furthest from western languages that you can go. Nevertheless, it's a great thing to have.
While Japanese is really a pain in the ass with its three scripts (bloody Kanji driving me insane while learning it) Korean actually isn't that bad. Unlike Japanese and Chinese the writing is pretty easy to learn for a Westerner.
It does include a couple of hanja but usually just uses the hangul alphabet. Which is just consonants and vowels... like Western languages.
If you truly want "pain in the ass", go for Japanese. Even Chinese seems easy in comparison with its simplified characters.
Knowing eight languages shouldn't be just a reason for admiration, I'd say it should be a nerd's fantasy.
Although I can only have a proper conversation in 5 or 6 languages (Italian, German, French, English, Spanish, and Japanese (unless the topic's too complicated). I never really tried actually having a conversation in Latin or Old Greek.