I know that there are tales of burghers and other commoners getting rich through trade. Was it unknown for villeins to climb the social scale?
I share everyone else's concerns about the assumptions embedded in this question, but I also think this is the kind of thing that H:SE is built for - to give the context to undermine flawed assumptions.
A villein is neither slave nor free; legally bound to the land, and to the feudal overlord. This is a society that measures wealth in land, and the villein by definition owns no land, so by definition it is not possible for a villein to be rich. (to accumulate wealth would mean to own land, which would mean that you are not a villein).
The economy is based on land rent and food rent - in the modern era we conceive of wealth as money, but the medieval economy didn't have much money. Debts and dues were settled in food and goods. There is effectively no way to accumulate food, and there are no coins to trade food for coin. Technically I suppose you could engage in a form of capital deepening, but if you started to accrue tools to increase your production, your feudal overlord would increase your rent. Given that the feudal overlord had full power of justice over you, the rent increase was automatically legal and supported by the state.
Is it, perhaps, worth also mentioning the Medieval belief in the 'Scala Natura' which was held to maintain that the class structure of society was ordained by God? @sempaiscuba
Finally, OP mentions Jews - I'd be very surprised if a Jew could be a villein. While there may have been edge cases, rural medieval society was not tolerant of diversity. Judaism is (in the general case) largely urban. (Keeping Kosher demands a community). I may be dead wrong on this, but my discussions with my Jewish friends have led me to believe that "Jewish villein" is statistically very improbable.
Mr. Geerkens is correct that Russian peasant life includes Jews, but I think that they were serfs, not villeins, and therefore even less likely to become rich. I think the chance of a Russian Jewish peasant gaining wealth is effectively zero.
I'm less convinced that the distinction between money and specie is relevant to the question. The question asks whether a villein ever got rich - "rich" would have been measured in land, not in specie. I think that the definition of "rich" is really important here. Certainly there were peasants whose peers perceived them as "rich" - relative wealth, having a better crop yield than a neighbor, or a better harvest, or whatever. But objectively "rich"? That has to do with property rights and social institutions that recognize a person's ability to control wealth and direct power. And for a villein to control wealth or direct power greater than his feudal overlord would have been such a threat to social order as to have merited drastic action. Kind of like if a nine year old were wandering down the street with a machine gun. Everyone would take action to stop that as a danger to publi order. Villeins with wealth would be like nine year olds with machine guns.
Extortion. Pure and simple.
You're a strong man with a retinue. What do you do? You set up a toll somewhere. Preferably a spot where people have to pass. Such as a fort in a river, or on a defensible busy crossroad. Fortify it, as passers by won't be much amused. Now you have money in the bank. Everyone passing your toll will have to cough up. Not much chance of them being able to storm your fortress.
That's what Dirk III did in Dordrecht. He set up a toll booth plus castle on the Merwede river (tributary of the Rhine river). Merchants had to pay a toll to pass. They weren't amused, and complained to the emperor. It took a while, but the emperor send an army to remove the toll. That imperial army was defeated. And Dirk collected happily ever after. Many nobles along the Rhine and other rivers ran a similar scam.
With regard to Jews lending money, that's a bit more complicated. Christians weren't allowed to levy interest on loans. Consequently nobody lent money. Until someone got the bright idea to force Jews to go into the money lending business. It's a bit like all gays being hairdressers or artists. Gays aren't all hairdressers or artists, but those were among the few professions open to them.
I think your mean the modern meaning of villein: villain. A villein was a poor peasant who was tied to the land of his lord. Those guys didn't get rich. They had no time for it, they had to work - hard. Some of course looked for criminal ways to get rich, but medieval justice was harsh if they got caught. Otherwise they had very little opportunity to get rich in one generation.
Their lucky break came when the Black Death ravaged Europe. If they survived, that is. The Black Death created a huge labor shortage. If the lord wanted them to keep working for him, he had to cough up for it. If not, plenty of better jobs elsewhere.
Maybe Ivaylo of Bulgaria?
According to some sources Ivaylo began his life living humbly and herded swine for payment. Other sources indicate he was a peasant farmer with no land of his own.
This caused an alliance between Ivaylo and Maria Kantakouzena, and Ivaylo married the widowed empress and was recognized as Bulgarian emperor in 1278, without deposing or disinheriting her minor son. Michael Asen III.