I was just thinking about how the Greek pantheon of gods supposedly lived on Mt. Olympus and about that mythological story of Bellerophon, who tried to climb Mt. Olympus with Pegasus. So I was just wondering if any ancient Greek (and by ancient Greek I'm thinking about Greek people before they converted to Christianity) actually attempted to climb the mountain.
To update my older answer to this question...
We should probably begin with a more basic question: did the Greeks really believe that their gods lived on Olympus?
The answer is more complicated than you might think.
Even at the beginnings of Greek literature, Olympus was not, or not just, an impressively cloud-capped mountain in Thessaly. In the Iliad, Olympos is both an actual peak (with epithets like "snowy" and "craggy") and a metonym for the heavens. At times, Homer's Olympus is clearly conceived as something more than a physical mountain, as when Zeus tells the other gods:
"If you tied a chain of gold to the sky, and all of you, gods and goddesses, took hold, you could not drag Zeus the High Counselor to earth with all your efforts. But if I determined to pull with a will, I could haul up land and sea, then loop the chain round a peak of Olympus, and leave them dangling in space. By that much am I greater than gods and men." (Iliad 8.19-26)
This dual conception of Olympus as both physical peak and heavenly realm continues throughout Greek (and later Latin) literature. The discrepancies between these conceptions are clear in the mythological compendium known as Apollodorus' Bibliotheca (probably written in the second century CE). In a myth about twin giants who attempted to storm the homes of the gods, the author notes:
"When they were nine years old and measured eighteen feet across by fifty four feet tall, they decided to fight the gods. So they set Mount Ossa on top of Mount Olympus, and then placed Mount Pelion on top of Ossa, threatening by means of these mountains to climb up to the sky" (1.53)
Here, at least, there is a clear distinction between Olympus and the home of the gods. The distinction in even clearer in Lucian's Icaromennipus, a satirical second-century text about a man who decides to fly to the home of the gods. Mennipus (the protagonist) doesn't bother with Olympus; he sets sail directly into the sky, and figures that the gods live very far off indeed. To quote his calculations:
"Let me see, now. First stage, Earth to Moon, 350 miles. Second stage, up to the Sun, 500 leagues. Then the third, to the actual Heaven and Zeus's citadel, might be put at a day's journey for an eagle in light marching order..."
By the time Lucian wrote, the educated elite, at least, tended to think that the gods were everywhere (the Stoic idea), nowhere (the Epicurean idea), or scattered among the stars (the Platonic idea). Calling the home of the gods "Olympus" had by then become little more than a convention.
So - having determined that many Greeks (at least in the post-classical era) probably didn't believe that the gods actually lived on Olympus - did anybody climb the mountain?
We don't know whether any Ancient Greek ascended to the highest summit of Olympus. We do know, however, that hundreds of Greeks routinely climbed to the sub-peak now called Hagios Antonios, about a mile away from (and only 300 feet lower than) the summit. There, from the third century BCE to the fifth century CE, offerings were made at an altar of Zeus.
From Hagios Antonios, there is a clear view of the craggy summit. It would have been obvious to all those who sacrificed at Zeus' altar that there were no gilded palaces or sunbathing gods on top. There seems, however, to have been a lingering sense that Olympus was a special place. According to Solinus, writing in the third century CE:
"The things that are to be seen at Olympus show that Homer did not celebrate it rashly. First, it rises so high, with a preeminent peak, that the inhabitants call the top of it heaven. On the summit is an altar dedicated to Zeus. If burned offerings of entrails are brought to it, they are neither blown off by windy breath nor washed away by rain, but as the year rolls on, whatever is left there is discovered unchanged; what is consecrated to the god triumphs over time and the corruption of the air. Letters written in the ashes remain until the next year’s ceremony." (8.6)
In the post-classical Greek imagination, the gods may not have lived on Olympus. But nor were they far away.
More can always be said on the matter, as no answer is ever 'final'; further answers are more than welcome from anyone able to contribute.
For the meantime, OP, you may be interested in the following previous answers on the matter of That Greek Mountain:
The ancient Greeks did climb Mount Olympus. By late antiquity, regular journeys up the mountain seem to have become incorporated into the ancient Greek religion.
Augustine of Hippo & Plutarch both specifically mentioned that there were regular pilgrimages up the mountain by priests.
A fragment from one of the last writings of Plutarch, preserved via quotation by John Philoponus in his Commentary on Aristotle's Meteorology, records that Greek priests would regularly climb to the summit of Mount Olympus & find inscriptions that they had left up there undisturbed every time that they returned.
The fragment from Plutarch reads as follows, as translated by Harry Sandbach for the Loeb Classical Library: ""For people who have placed on top of some mountains, or have left it behind after sacrifices there, have when investigating many years later found that it was still lying as they left it." ...Plutarch reports that letters, too, remained from one ascent of the priests to the next on Olympos, in Makedonia.".
Augustine of Hippo wrote something very similar in his The Literal Meaning of Genesis, but seems to have misinterpreted; Augustine claimed that the priests would write things in the dust on Mount Olympus & return years later to find their markings undisturbed by the wind. Plutarch, who is the earliest source, however, seems to have described inscriptions carved in stone - not writing in the dust.
Additionally, there is also ample archaeological evidence of ancient Greek presence on Mount Olympos. Archaeologists from the Aristotle University from Thessaloniki have found ancient Greek pottery fragments near the tops of some peaks, a coin dated to the 3rd century BCE, & also the remains of burnt sacrifices & other offerings that were left for the gods atop the Agios Antonios peak. They have even found several ancient inscriptions atop the mountain like the ones described by Plutarch. 2 of these inscriptions specifically mention "Olympian Zeus".
There is overwhelming evidence that the ancient Greeks not only definitely climbed Mount Olympus, but that, at least in late antiquity, this was a relatively common occurrence & that people even sometimes offered sacrifices to the gods up there.
It is likely that the ancient Greeks who lived close enough to Mount Olympus to climb it did not think that the gods literally physically lived on top of the mountain in exactly the same way that humans might live there.
Certainly, some people in ancient Greece did believe that the gods literally lived atop Mount Olympus in ornate palaces, eating ambrosia & drinking nectar just like they are described as doing in the Homeric epics, but the ones who were climbing Mount Olympus evidently did not believe this. It is likely that many of 'em thought of Mount Olympus as more of a metaphorical home for the gods, rather than their actual literal abode.
A specific example would be the famous Olympian Polydamas of Skoutoussa. While he did have very unlikely feats ascribed to him, he is not a mythic hero in the way that somebody like Heracles is, and he is associated with historical events (notably the 93rd Iliad from 408 BC)
Anyway, around 500 years after when Polydamas lived, Pausanius described him climbing mount olympus and killing lions with his bare hands. No presumption of divine presence is really made in this account.
http://perseus.uchicago.edu/perseus-cgi/citequery3.pl?dbname=GreekTexts&getid=1&query=Paus.%206.5.5