Assuming you're a classical music aficionado and/or trying to show the Australian girl from the hostel how 'cultured' you are, then you'll be pleased with Amsterdam's many free orchestra performances. The Dutch National Opera & Ballet and Concertegebouw Orchestra hold complimentary lunchtime concerts on Tuesdays and Wednesdays, respectively. There’s also the Muziekgebouw aan t’ IJ, a slick-looking concert hall where conservatoire students play gratis on the last Thursday of every month.
The Acropolis is incredible, but it sucks for two reasons: 1) summer admission is going up to $22, and: 2) throngs of tourists crowding your Parthenon selfies. So head down to Filopappou Hill for a panoramic view of Athens, the Acropolis, and even the Aegean Sea. Walking up, there are also a couple more cool (and yes, free) attractions to be found, like a 16th-century church and the prison where Socrates is thought to have died.