The United States is often referred to as a "class-less" society where anyone can rise to any height. Leaving aside whether this is actually the case or not, when and how did this idea originate? Was it part of the Enlightenment ideas of the founding generation, or was it something that came along later?
The United States was in the "new world." As such, it didn't START with many of the class structures common to European societies.
As such, it was regarded as a good "testing ground" for theories of a classless society stemming from the Enlightenment. The "founding generation," even though heavily tilted toward the upper class, was greatly influenced by these ideas, and wrote them into the Declaration of Independence, the Constitution, and other founding documents.
America has never been a fully "classless" society. What HAS been true is that many of the class barriers that stood in people's ways in Europe didn't operate particularly well in the U.S. In an achieving society, the ethos has been to promote a meritorious individual from the "wrong" background over a mediocre one from a good background.
Western European countries have largely adopted "American" ideas of a democratic meritocracy, to their benefit. Some of them now arguably have more social mobility than the U.S.
But the U.S. will probably long maintain its reputation as the site of the global "lab experiment" in equality.
The last "class" concept I can think of the U.S. having would be segregation. I believe it was the "everyone is equal" movement (The whole "sitting at the front of the bus" thing) that lead to our current class-less "everyone is equal" state.
Even if somebody can rise to height, does not make a society classless. Class is not an sealed set of people: people always can move from one class to another. You possibly confuse class with a social estate or caste the two being more closed divisions of society without easy ways to change.
What distinguishes class (by Marx) is the possession of the means of production. That means if a society has those who possesses means of production and those who do not, the society is not classless.
Income also is not a distinguishing criterion: a hired manager can have higher income than a farmer, but still he belongs to a different class.
Not sure I understand well this question: Jefferson writes in the Declaration of Independence:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
"The answer is staring you in the face."
Why is the founding document of the United States, often memorized by school children, traditionally read in public on Independence Day, not sufficient to answer this question? It is the document upon which everything rests.
By law, a Constitutional Convention can be convened and a new constitution of the USA can be drafted and ratified. But the United States as a nation remains, as per our Declaration, and therein is stated quite clearly the idea of a "classless society".
(There is much more here - but alas, I have not the time to elaborate at the moment or cite some of the original sources. Perhaps in time - see the wiki on the Declaration for a starting point.)
Gordon Wood's The Radicalism of the American Revolution contains a long discussion of this concept. (personal opinion, the discussion extends longer than useful, and seems focused on responding to an argument that isn't in the book).
I believe the question is founded on (one or more) false assumption; "The United States is a class-less society" is a conclusion, not a pre-existing idea. (others have discussed ad nauseum the imprecision involved in "class-less"). During the 1740's to 1780's, the institution of "class" was dysfunctional in the united states. (Here, I reject the marxist definition as lacking in utility; I refer to the more contemporary notion of gentry class vs commoner). The things that made that institution work in Britain (including the presence and active intervention of a King, a populist Tory party, and a strongly established Church of England) were absent in America.
One could argue that classlessness is an artifact of Republicanism of the period, but even Republicanism was fundamentally oligarchic. (one of the reasons for the failure of the Articles of Confederation was the "licentiousness" of the American populace. Writers of the period lamented the notion that craftsmen and tradesmen and merchants were involving themselves in politics, which was unacceptable). But Republicanism within the British constitution (such as it was at the time) was different from Republicanism in the Netherlands or in Switzerland - I think the argument very quickly drowns under the difficulty of defining Republicanism and Classlessness.