If we are talking purely linguistic features, I would study a Semitic language like Arabic or Hebrew because of their fascinating way of extending vocabulary by taking root words (typically a series of three consonants) and inserting various sets of vowels between the consonants to form related words and parts of speech. That's a brilliantly economical way to produce new words, and the morphology is so different from the western European languages I've studied that it fires up my interest and imagination.
For the tremendous agglutination and long series of infixes, I would choose Turkish. It would also be cool to speak a language that can be 70% understood thousands of miles away in Turkmenistan, but maybe that counts as an external factor.
For variety, I would choose one highly inflected language and one highly isolating language. My choices would be Russian and Mandarin Chinese, but for external reasons (the literature, the global reach, the cultures). I know that Hungarian has more cases than Russian, but that degree of inflection just becomes pure torture to learn. Seven cases is enough for me! (I loved learning German with its four cases. I especially loved that magic moment when my brain suddenly flipped and I woke up one day, after months of struggle, mysteriously able to get all the sentences out easily with the right case endings on each noun without having to consciously think about it anymore. Achieving automaticity in a highly inflected language brings its own "high.")
After reading Nicholas Ostler's Empires of the Word, I'm dying to learn ancient Greek, mostly because it was the lingua franca of such a vast swath of the world for centuries, but also because of "outside" factors like the literature and philosophy, which would draw me to both the Attic and Koine dialects/varieties. Given that rationale, it's strange that Latin doesn't interest me as much, perhaps because of its familiarity to me as a speaker of English and French (and perpetual student of Spanish). (I also dislike all those declensions in Latin, but Greek has plenty of difficulties of its own, so I realize I'm being inconsistent.)
I doubt I'll live long enough to get around to even a fraction of this (but isn't the number of years that I'll be alive an "outside" factor?)
Please, someone give me a million dollars so that I can quit working and spend all my time studying languages!EDIT: Added parenthetical snark to next-to-last sentence.
