We all know that the ancient Greeks believed that their gods lived on top of the mountain Olympus (and some of them in other places like the Underworld). But Olympus - they knew where it was (between Macedon and Thessaly) and also that mountain was never higher than 3.000 meters, so it was always pretty well climbable even in former times.
Is there any evidence of the ancient Greeks climbing Mount Olympus?
The answer is probably Yes, some Greeks had visited the peak around 2900 meters above the sea level. Average people who are fit can easily do it, at least from one side.
But it's important to realize that they didn't expect to meet gods there. According to the Greek teaching, the divine and human spheres only overlapped but they were not identical. One could "feel gods" at the peak but not really see them.
Moreover, the peak of Mt Olympus wasn't the only one where gods were said to "live". For example, the river Achelous was both the "body of flowing water" and god named Achelous, the boss of all other rivers etc. Achelous was thought to be as real and as anthropomorphic as Heracles or anyone else – Achelous actually appears on a painting along with Heracles. And Greeks could surely see the whole river of the same name. But they didn't expect to meet the god near the river.
The reality wasn't "identical" to the divine entities in their religious legends. It was just an incomplete projection of it. This type of thinking with the "layers that are hidden from human senses" may also be seen e.g. in Plato's allegory of the cave.
It is impossible to ascertain for sure.
Eleven gods of the Greek major pantheon all lived in various named corners of the mountain. Even Zeus resided on only the second-highest of the many peaks, with the highest, Mytikas, reserved as their meeting pace.) This suggests that the ancient Greeks were familiar with the terrain of the mountain.
The summit height is less than the 10,000 feet altitude where an oxygen supply might be advised, and while technical competency (and helmets) are necessary for a few of the many peaks, ropes are not necessary for any of them. Several of the lesser peaks can be hiked up with only good lungs and strong legs. At least 10,000 climbers/hikers ascend at least one of the peaks every year.
However, the first known ascent of (the peak of) Mt. Olympus was only in 1913.
Just remember - if one is a god, making yourself either scarce or invisible when an inquisitive human comes nosing around is a trivial feat.
You are assuming that the Ancient Greeks believed that their Gods were humanly visible, which is unlikely. Therefore, whether or not they climbed Mount Olympus, the Ancient Greeks would not have expected to visibly see their Gods.
From Classical Myths by Barry B. Powell (a textbook often used in undergrad classics courses):
In other myths, the setting is an obscure place: the underworld, which no one in the real world ever visited; Mount Olympus, which really exists but in myth is the home of the gods; or Crete of a very long time ago.
Also:
The events of divine myth usually take place in a world before or outside the present order where time and space often have different meanings from those familiar to human beings.
Basically, their Gods were there, but not humanly visible. That makes sense.
As to whether or not the Ancient Greeks ever climbed Mount Olympus, some sources say that they did and left offerings there for their Gods, later discovered via archaeological excavation.
The first men to reach the summit (as far as we know) were Frédéric Boissonnas and Daniel Baud-Bovy in 1913. But from the peak Aghios Antonios you get a view of the two highest peaks.
I believe that regardless to common beliefs that there are and where always human beings who want/ed to experience as much as they can/could. So even if there was no historical evidence about it, I find it very unlikely that nobody climbed mount Olympus.
And I do not believe that it was the light headedness that they felt as the presence of the gods. It would not explain why they felt this presence elsewhere too.
But I think that if somebody is a strong believer and has a higher mental imagination and perception, that both of them enforce each other at special locations. With higher mental imagination I also mean the ability to highly abstract thoughts, feelings, emotions, etc. or reduce them to their essence. For example the ability to think about something and entirely capture it without thinking of the describing words or images.
It is comparable to intuitively understanding a program or a mathematical problem without being able to describe it, just in another domain.
I'm thinking about Ötzi: https://en.wikipedia.org/wiki/Ötzi He was found mummified at a height of 3,210 metres (10,530 ft) in the Alps. So it is certainly possible that the greeks could have reached the Olympus summit. But I also suspect that if they did, at least someone of them would probably die. At least if it were a common practice to "visit the gods". Sooner or later, someone would find himself in bad weather and freeze to death. Ötzi shows us how well a body of one visitor could be preserved even after many thousands of years. Therefore I suppose it is unlikely the greeks climbed Olympus.
