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There aren't any staircases known to be that old. There's an interesting anecdote in Herodotus though. I like Herodotus' sense of perspective and his willingness and ability to think for himself. I added the boldface in the quote for emphasis.

The Histories
by Herodotus
tr by George Rawlinson
Book II -- Euterpe
The Egyptians... told me that the first man who ruled over Egypt was Min, and that in his time all Egypt, except the Thebaic canton, was a marsh, none of the land below Lake Moeris then showing itself above the surface of the water. This is a distance of seven days' sail from the sea up the river... Now if the Nile should choose to divert his waters from their present bed into this Arabian gulf, what is there to hinder it from being filled up by the stream within, at the utmost, twenty thousand years? For my part, I think it would be filled in half the time. How then should not a gulf, even of much greater size, have been filled up in the ages that passed before I was born, by a river that is at once so large and so given to working changes?

...One fact which I learnt of the priests is to me a strong evidence of the origin of the country. They said that when Moeris was king, the Nile overflowed all Egypt below Memphis, as soon as it rose so little as eight cubits. Now Moeris had not been dead 900 years at the time when I heard this of the priests; yet at the present day, unless the river rise sixteen, or, at the very least, fifteen cubits, it does not overflow the lands. It seems to me, therefore, that if the land goes on rising and growing at this rate, the Egyptians who dwell below Lake Moeris, in the Delta (as it is called) and elsewhere, will one day, by the stoppage of the inundations, suffer permanently the fate which they told me they expected would some time or other befall the Greeks.

and for that matter:

The Histories
by Herodotus
tr by George Rawlinson
Book II -- Euterpe
Thus far I have spoken on the authority of the Egyptians and their priests. They declare that from their first king to this last-mentioned monarch, the priest of Vulcan, was a period of three hundred and forty-one generations; such, at least, they say, was the number both of their kings, and of their high-priests, during this interval. Now three hundred generations of men make ten thousand years, three generations filling up the century; and the remaining forty-one generations make thirteen hundred and forty years. Thus the whole number of years is eleven thousand, three hundred and forty; in which entire space, they said, no god had ever appeared in a human form; nothing of this kind had happened either under the former or under the later Egyptian kings. The sun, however, had within this period of time, on four several occasions, moved from his wonted course, twice rising where he now sets, and twice setting where he now rises. Egypt was in no degree affected by these changes; the productions of the land, and of the river, remained the same; nor was there anything unusual either in the diseases or the deaths.

...When Hecataeus, in giving his genealogy, mentioned a god as his sixteenth ancestor, the priests opposed their genealogy to his, going through this list, and refusing to allow that any man was ever born of a god. Their colossal figures were each, they said, a Piromis, born of a Piromis, and the number of them was three hundred and forty-five; through the whole series Piromis followed Piromis, and the line did not run up either to a god or a hero. The word Piromis may be rendered "gentleman."