Hadrian's library is one. Of the easiest sites to find in Athens, and a great way to slow down the pace of a frantic day of touring around the city. Only 20 meters from Monastiraki square in the heart of the old neighborhood. To the untrained eye the site appears to be fairly basic, with the most notable features being the giant pillars found at the front entrance. Other than that there are many large unremarkable walls around it and a few beautiful arches and ruins in the middle. It's not a large site in area compared to the Acropolis or the Agora, but there's still lots to see.
The beauty of this site is how quite it is, considering it's surrounded by some of the most hectic parts of Athens. A mere 20 meters from the entrance is one of the busiest squares in the area, but you forget all about it wandering around. The site isn't full of particularly remarkable buildings or structures, but lying around are wonderful little treasures that you can discover yourself. Bits of mosaic floor, grinding stones, and a few pillars are hidden all around, but you have to keep your eyes open to find them.
This is a wonderful site to explore on your own, it's often not very busy, and it's a great escape for a few minutes.