The consequences of diglossia, which persist even if diglossia itself does not—including the trainwreck of Modern Greek phonology from all the spelling pronunciations from Ancient Greek, the lexical and morphological doublets, and the all-round linguistic insecurity.
The survival of archaisms in Indo-European, including the middle voice (semantically), the vocative, and the three genders
As Joachim said, the survivals and reorientations from Ancient Greek. I’ve often been curious what Modern Greek looks like to classicists; the best answer I’ve heard is “drunk”.
The dialectal diversity (though Greek is hardly unique in that.)