Greek

Greek is the longest continuously documented Indo-European language — 13 million speakers in Greece and Cyprus, plus a global diaspora that traces straight back to Homer. It's the parent of half the technical vocabulary in English (philosophy, democracy, telephone, hydrogen) and the vehicle of one of the most influential literary, philosophical and scientific traditions on earth. Even three months of Greek study makes you read English differently.

For English speakers, Greek is moderately challenging in fascinating ways. The alphabet looks alien but is mostly familiar — half of it is the same letters used in maths and engineering, just with different sound values. The grammar has cases (four), genders (three) and a verb system with two distinct stems for completed and ongoing actions, which feels strange at first and then unexpectedly elegant. Best of all, the language opens up a country whose hospitality (philoxenia, literally "love of strangers") is built into the grammar itself.