Anargyri’s name day is November 1, the feast of Kosmas and Damian (Asian).
Churches that follow the Old (Julian) Calendar keep the same feast on November 14 of the civil calendar. See everything commemorated on November 1.
The saint behind the name

Kosmas and Damian (Asian)
November 1, Saint, Κοσμάς και Δαμιανός
Saints Kosmas and Damian of Asia Minor were brothers and physicians, raised in the faith by their pious mother Theodota. They treated the sick without ever accepting payment, asking only that those they healed believe in Christ, and for this the Church calls them Unmercenaries. They reposed in peace after lives given wholly to God and neighbor.
Name days: Kosmas, Cosmas, Damianos, Damian, Argyris, Argiris, Anargyros, Argyro, Argiro, Anargyri, Damiani
Also celebrating on November 1
These names share the feast, many of them other forms of the same name:
Kosmas, Cosmas, Damianos, Damian, Argyris, Argiris, Anargyros, Argyro, Argiro, Damiani
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.