Since "history" indeed meant in Ancient Greece "inquiry", it means that your curiosity being answered and satisfied was "history" but not as the record of facts but as the record of plausible (in their minds) but non-demonstrable by other witnesses facts.
So I wanted to know, what is the oldest recorded act in the history of what we know as Ancient Greece that we can state as fact?
Seriously, what is it? that there was a war between Eretria and Chalcis? It certainly got exaggerated beyond plausibility for different literary motifs (like Thukydides wanting to put an analogue to the Peloponnesian war as the Median War was analogue to the supposed Troian War)
The First sacred war is nuts. Beyond impossible given that neither Herodotos nor Thukydides spoke of it.
The rebellion of Kylon in Athens? The "First" Olympic Games? The founding of the earliest colonies in Italy and Sikilia? Which one is that you kind and excellent historians can state without any doubt as fact? I start to feel even if it is something, some fact or event, stated by both Herodotos and Thukydides, it would not mean they would not be cheated by the sources. Certainly Herodotos heard too much crap of Persia to be truthful (Rest in Peace even if also in Pieces Darius noble and poorly framed son of Xerxes I, and fuck you Artaxerxes I, even the second brother had to go because of you)
As you know well enough, "without any doubt" is a very tall order for Ancient history. Almost any fact about Antiquity can be doubted for one reason or another. Events attested by multiple, independent sources are rare; the ways we use to date things (objects, migrations, wars, lives) are often contingent on the assumption that at least some of these sources must be accurate, or that our own rough estimates based on style and stratification are correct enough to serve. Chronologies vary in accuracy based on what sources happen to be available for a particular period. This also means that the degree of certainty of our knowledge does not increase in a linear way as we get closer to our time; we know much less about events in Greece during the 3rd century BC than we do about events of the late 5th. We have to make do with what we have. It is therefore not particularly helpful to insist on facts that are completely beyond doubt. We don't know nearly as much as we'd like about the Ancient world; we'd know practically nothing if we restricted ourselves to what we know "without any doubt".
This is particularly true for Early Archaic Greece, about which the Greeks claimed to know various things, but clearly actually didn't. Most of the things claimed in later traditions have been exposed as myth-making, retrospective impositions, genealogical invention, or simply unfalsifiable speculation. The Dorian migration, the reforms of Lykourgos, the Lelantine War, the First Sacred War, the First Messenian War, the start of the Olympic Games in 776, the battle of Hysiai in 669 - all these things have proven impossible to confirm either archaeologically or otherwise, and have consequently been doubted or at least downdated significantly by modern scholars. We're free to assume they are all real events, but we can never be certain. It is much safer to say that the Greeks believed these things happened, and recounted them as historical, despite the fact that, even in their own day, they had become impossible to recover. Often their reasons for preserving these stories had more to do with their sense of community, their political claims, or the genre in which they were writing, than with verifiable historical fact.
It's worth asking yourself, then, what you're actually asking for here. What evidence do we need in order to state something "without any doubt as fact"? Are we looking for material evidence? Confirmation by independent sources? Secure dates? Depending on our standards for what amounts to a fact "without any doubt", we might end up in very different places.
For example, we could say that we know "without any doubt" that there was a time when much of mainland Greece was ruled from palatial centres that lost their power and prominence around the 12th century BC. We know this because the ruins of those palatial centres are still there, and they show lots of evidence (artistic styles, Linear B writing) that they were in use during a time well before "classical" Greek civilisation. However, we cannot precisely date the collapse of these palaces, and we don't actually know what caused it. The Greeks themselves are of no help to us here, because they did not remember.
For another example, the earliest event for which we have a mostly uncontroversial date is the foundation of Naxos on Sicily in 735/4 BC. The date is derived from Thucydides, who seems to have gotten it from a lost Syracusan chronicler. Archaeology confirms the appearance of a settlement around that time, and shows that whoever lived there used Greek pottery. However, we can never be sure if the arrival of Greek pottery means the arrival of Greek people; most settlements in the area show intermingling of local and imported artifacts. In addition, the styles of the pottery suggest that Thucydides may have been a few decades off. It has been suggested that the dating system he used was based simply on multiplying a known number of generations by a set number of years (35 or 40) in order to arrive at an approximate date. So, while we know that a settlement grew at Naxos in the second half of the 8th century BC, we cannot say "without any doubt" that this was indeed the city founded by the Chalkidians led by Thoukles, or that it was founded specifically in 735/4 BC.
Casting doubt on what we know for certain is an important part of the study of Ancient history. It is the main reason why our understanding of this period is always shifting, and why there is always more to be said on a period that ended thousands of years ago. It is not our goal merely to discover what facts are known "without any doubt", but rather to see if we can construct narratives in which what we think we know makes sense, and in which the uncertainty of facts is not a critical weakness.