...is a long-standing tolerance and acceptance of tax evasion.
As this article at Bloomberg explains:

Apostolis Rigas took his Opel sedan for a 220-euro ($354) service at a repair shop in northern Athens. When he asked for a receipt, the price jumped to 260 euros as his mechanic would have to declare the income and pay tax.
“There’s no taboo about this,” the 23-year-old student said in a Feb. 2 interview. “Tax evasion helps support families, but it’s not good for the Greek state.”

Leave it to this economist to spell out the elephant in the room:

“What distinguishes Greece from the rest of the pack is the extent of tax evasion,” said Michael Massourakis, chief economist at Athens-based Alpha Bank, the country’s third biggest-lender, in a Feb. 5 telephone interview. “If you don’t attack tax evasion you don’t have the moral authority to cut spending.”

My own personal anecdotal example: when I went on vacation in the Greek Islands, it did not take long before I noticed that many buildings seemed to be incomplete, with the base pillars for an additional floor disturbing a flat roof for the existing 1 or 2 floors. However, it was also clear that many of these buildings were decades old.
When I asked the owner of a bar we frequented on vacation about this, he explained that it was a tax dodge. Once a building is complete, a tax is levied on the building; however the tax is not levied if the building is not completed. House owners would therefore have an architect draw up plans for an additional floor on a house before the house was built. When the house was built they would have the builder put the base supports in place, so that they would be able to report the house as incomplete. Voila! No tax liability. My travels in Crete and Skiathos showed that this dodge is endemic; most buildings are "designed" to be perpetually incomplete.

The problem with cracking down on tax evasion is that it is a centuries-old tradition:

...public anger in Greece taps into a tradition of tax evasion-as-protest against nearly four centuries of rule by the Ottoman Turks that ended with Greek independence in 1829, Massourakis said. Even for those who pay, colluding with tax-dodging of taxi drivers and bar-owners is still considered a form of solidarity.
“If this was a friend of mine he wouldn’t give me a receipt and I wouldn’t ask,” Rigas said. “I’m not so sure they’ll succeed.”

The massive budget deficit that has pushed Greece to the edge of bankruptcy is challenging enough, but a bigger challenge is how to modify a national culture that sees tax evasion as normal and even honorable.