Mount Athos is the site of a large number of monasteries that have been more or less the spiritual heart of Eastern Orthodoxy for at least 1,200 years, if not a little longer – with a break between the fall of Constantinople in 1453 and the restored communication between Greece and Russia under the Romanovs (The Way of a Pilgrim and the sayings of St. Seraphim of Sarov are both products of that rediscovery, and Athonite asceticism was embraced by a lot of Christians frustrated with Imperial attempts to dominate and modernize the Church and Russia in general).
It was also the most notable holdout against the Iconoclasts during that controversy, which was the last major dispute in the East to be solved by an Ecumenical Council. Given the importance of icons to daily Orthodox practice and their popularity among even illiterate laity – in contrast to other disputes about the nature of Jesus that are often over the heads of people who haven’t studied Christology in depth – the Athonite monks are respected as defenders of Orthodox belief who are above the influence of temporal politics (Iconclasm itself being largely a reaction to Byzantine defeats by Muslim armies who pointed to iconography as proof of religious corruption in Byzantium).
Greece treats it as something of a protectorate, and one stipulation of their entrance to the EU is that it retains control of any movement of people or goods within its borders. All the permanent residents are male monks, many of whom are not Greek, so for the most part the only people allowed on the peninsula are male pilgrims, in accordance with the monastic rules that have been around for rather a while now.
As an aside, at least some of the monasteries there are still flying the flag of the late Byzantine Empire; I have no idea if they just kept them up during Ottoman rule or brought them back out when Mt. Athos became part of Greece.
The monasteries themselves are technically under the jurisdiction of the Ecumenical Patriarch, not the Greek Orthodox Church, and in recognition of that as well as its unique place in Greek history, it’s given a special status.