It is the oldest functioning monastery anywhere. There are 25 monks there today, servicing the smallest diocese in the world.
The monks are all Greek, with one exception: Father Justin, a converted Baptist from El Paso, Texas.
Father Justin, the chief librarian, oversees an incomparable collection of ancient manuscripts. "Most of these date from the 10th to the 14 century," he explained.
He showed us the monastery's collection of Byzantine icons, the largest and oldest collection of icons in the world.
Then he took us to what the patriarch really wanted us to see: a little-known letter written by the Prophet Mohammed, almost 1,400 years ago, signed and sealed with his hand print, offering protection and religious freedom to the Christians of the monastery.
"These are precedents from Mohammed himself for toleration and peace among people of differing faiths," Father Justin explained.
The patriarch then brought us back to the 21st century and Turkey, to his own back yard. He took us for a ride on an island off of Istanbul in a carriage, with a police escort. The patriarch wanted to show us that Mohammed's message of tolerance has not been received by the Turkish authorities.
His prime example is the Halki School of Theology, the only Greek Orthodox seminary in Turkey. Empty and abandoned, there are no priest and no prayers.
The Halki was closed down by Turkish authorities after passage of a law banning private higher education. That was back in 1971.
"It's a pity and a shame. It's a crime to keep such a school closed, unused, for no reason. Why?" the patriarch asked.
"Reasons of state?" Simon asked.
"Reasons of state," the patriarch replied.
As a consequence, the church can't train new priests - potential new patriarchs who, under Turkish law, have to be born in Turkey. It's as if Rome closed down the College of Cardinals.