Visiting a museum is one of the most interesting things to do when travelling. Especially in Greece, with its numerous archaeological sites and museums, this cultural activity can be one of a kind. The good news are that the entrance fee in Greek museums and archaeological sites is cheap, but the best news are that you can find in Greece, and especially in Athens, free museums or museums with free admission days.
Every first Sunday of the month between November and March, all museums and archaeological sites in Greece are free. For those under 18 years, they are always free. If you visit Athens in the summertime and you can't take advantage of this free admission day, you should remember that the Acropolis ticket (12€) includes the Temple of Olympian Zeus, Ancient Agora (Athens), Roman Agora and Keramikos Archealogical site.
Benaki Museum, one of the best museums in Athens, is free on Thursdays.

Archaeological Museum Athens
However, there are in Athens museums that are always and for everyone free:
Kanellopoulos Museum: a wonderful cultural museum in a historic mansion in Plaka.
Museum of Greek Popular Instruments: it displays traditional instruments and costumes as well as recordings and other folk art items.
Epigraphical Museum: a significant collection of Greek ancient inscriptions on stone tablets.
Centre of Folk Art and Tradition: periodic exhibitions in a stunning mansion in Plaka.
Maria Callas Museum: a museum dedicated to Maria Callas and her voice.
Museum of Eleftherios Venizelos: in an independent building next to Eleftherias Park Art Centre, the museum hosts a permanent exhibition on the Greek statesman Eleftherios Venizelos. It features his personal collection of books and written material as well as the historical events happened in his era.
In Athens, you can find also many art galleries with exceptional exhibits of modern art, which are always free.