Rome is my area of expertise so I will give on major example if a very wealthy man with no political interests. Unfortunately most of these people are lost to history since without political power and just money most of their legacies are not known to us. One who we do know a lot about is Titus Atticus, a wealthy equestrian banker who was very good friends with Cicero thus we have quire a bit of information on him. He was wealthy enough that he could give Cicero 250,000 sesterces without much difficulty. And to have some idea on that having a million sesterces was enough to become a senator. Also there were no doubt many titans of industry because Roman laws forbid any of the larger traders from becoming Senators by having a rule mentioned in Livy that a senator could not have a ship larger then 7 tonnes.

Absolutely. Unfortunately, your question is framed with a very industrial mindset, or at least one that's projecting a late-medieval, 18th Century, and Victorian class structure back into antiquity. The class structure of most places in the ancient world didn't work that way. It wasn't really particularly cemented and there really wasn't very much to distinguish people from different classes except for very slightly different privileges and the preservation of different traditions. But there was actually a great deal of social mobility in antiquity, even in a society such as Rome's that preserved an oligarchic aristocracy. The nobiles at Rome had nothing to do with their heritage. Nobilis descends from a word that means "noteworthy," or "important"--in other words, the nobiles were the families that were important at any particular time, and they varied quite drastically often. True, the same few families tended to dominate the group, but that's not surprising and it's not to say that the roll of noble families didn't change often--in fact, it often changed within the same generation.

In Athens the class structure is quite easy to comprehend. Although there was a concept of an aristocracy that had carried over from the Archaic Period, it didn't necessarily correspond to the property classes. What's interesting is that of the four property classes, the wealthiest class was valued only just over twice as wealthy as the zeugitai, which was the hoplite class. The thetes, the lowest class, was anything below the zeugitai, but since the pentakosiomedimnoi, the highest class, were measured at anyone over 500 medimnoi (a medimnos is a measurement of volume used for grain) produced annually and the thetes were anyone under 200 medimnoi, there wasn't that large a wealth gap. At least in theory. But we know that there were almost no pentakosiomedimnoi, just a handful in the 5th Century, and that they were considered extremely wealthy. Compare that with the modern equivalent, which can have hundreds of times the wealth of the average income, and are unimaginably wealthier than those on the poverty line.

Anyway, the most famous man of wealth at Athens was Nicias. Recall that at Athens during the 5th Century there was no aristocracy. Now, Nicias had inherited most of his wealth, which he then invested, but where did his father get it from? Well there were several ways to do that in Athens. Landed wealth was pretty insignificant compared to what you could make by investing in merchants or hiring out gangs of slaves. Nicias' wealth was mostly in terms of the number of slaves that he owned, which was enormous, and he hired these slaves out to the state to work, mostly in the silver mines at Laurium. So his father probably invested in slaves or something like that. We have attested men who invested in merchant expeditions and made enormous amounts of money off of them, which required no land.

What about Rome? Well the Romans did have a noble class, although like I said this changes around a lot. The nobiles were generally the old patrician families, which could be added to but were rarely removed, and the notable equites. In theory there was a property requirement among the senators, although the censors often ignored this (one of the reasons Cato the Censor was considered so unjust was because he actually followed the law of the property requirements to the letter), but you couldn't be demoted from a patrician for something like that. Now the notable equites became notable for various reasons, often impressive military service (like Pompey or Marius) but also due to wealth, usually if that wealth served the state. Most equites were known for building their wealth not from the spoils of war but from trade, like Atticus. Cicero's father had risen to prominence due to his enormous wealth, and Cicero himself was able to secure his seat in the senate in part because of the means that such wealth gave him. And let's not forget Crassus. Crassus was another one of those equites who made a ton of money and got himself accepted into the ranks of the nobiles. Now, the gens Licinia to which Crassus belonged is somewhat tricky, since they were technically a plebeian gens (although later on most branches of the family had been promoted to equites), but it was among the most prominent ones, since it had held many of the earliest tribunates and was one of the first non-patrician families to obtain the consulship. However, there were three main branches and dozens of little ones, and while Crassus was the son of a distinguished senator his branch of the family was among those not accepted by the old landed aristocracy (like I said, aristocratic relations and classification in Roman society is the subject of countless books and studies which it's not possible to easily summarize). His position as an important political figure was not based on his father's name (after all, his two brothers are virtually unknown) but due to the wealth that he amassed, which allowed him to fund many notable statesmen. That wealth was not inherited from his father, because although they were equites and although his father was a distinguished military figure, the family actually had very moderate wealth (by the way, Crassus' reputation as the wealthiest man in Rome is an exaggeration that he and Caesar proliferated. As Badian pointed out 60 years ago, after Pompey returned from his wars in Asia the loot that he brought back with him and the income that he could make from the tribute he was exacting from the east made him many times wealthier than Crassus could ever dream of being--which was probably one of the reasons Crassus later marched on Parthia). Crassus largely built himself up on his own.

There's also the story of Trimalchio in Petronius' Satyricon. Not only is the character not a noble, he's not even Roman--in fact, he's an ex-slave. This is something that was common in both Greece and Rome, because since slaves could own property and were regularly freed after a certain amount of service by their masters at both Athens and at Rome you often found slaves and former slaves building up huge fortunes, especially because former slaves had not problems with entering professions that most Romans looked down on. These men were so common by Petronius' time that the character of Trimalchio is obviously intended to be a familiar type with the audience, reflecting common behavior (usually crude and unsophisticated) of this nouveau riche

I think the answer to this question is a little bit more complicated than what the other answers are giving. The problem is that Greece and especially Rome did not really have the sort of social or economic structure you are asking about. They most certainly had a hereditary elite class (and Greece's even justified itself through complex genealogies traced back to heroic times) but it wasn't one that maintained itself through a particular familial or "House" association. The Cornelli, for example, may have been a family with a very deep history dependent on landed wealth, but the justification for their high status was not their decent or bloodline but rather the offices they attained. It was a so-called "aristocracy of service" undergirt by republican ideology of civic duties, although we should be careful not to confuse this distinction in terms of self-representation with as actual greater degree of egalitarianism or "fairness" in the social order. In practice, the Cornelii could and would use their association with illustrious men of the past to their advantage, and of course elite in more overtly feudal society would position themselves through actions and deeds, but the self conception of the class was quite different. Rome and even more so Athens' class structure was very much unfair but it was also one with a degree of permeability.

In practical terms, this means that a position of elite status in Rome was one that needed to be constantly renewed. Even Catullus, the great poet of otium ("leisure", although it is a more complex concept than that) still dutifully worked his way through the series of Republican offices. The Principate changed this dynamic but did not destroy it, and what we see (although this is a characteristic that certainly begins with the Late Republic) is a great proliferation of ways to "renew" this status. Political office is of course a major one--the Emperors co-opted the system of republican magistracies and offices rather than doing away with them. They could also engage in euergetistic activities (donations, building projects, etc), literary and scholarly production and patronage, social engagement, etc.

So what does this have to do with money? Well, I think it is important to start with an understanding of who these elite were before we can understand how they managed to obtain the wealth that allowed them to engage in these "renewal" activities. Unfortunately, we do not possess anywhere close to the sorts of records that would allow us to say what the primary means of wealth were (although it was almost certainly agricultural, as is standard in pre-industrial economies), or more importantly, the sort of diversities present within the elite class in terms of how they made their wealth. But from comparative evidence and what scattered remains we do have, something that emerges is the presence of an "Imperial elite" who made their money through their relationship to the state. If you think of Medieval Europe as having a "feudal economy" (not that you should, of course) in which wealth was created and maintained through the socio-political domination of particular areas and the people within them, you can picture Rome as having an "Imperial economy" in which an enormous amount of money was flowing across the empire due to various imperial activities, and wealth was created and maintained through tapping into this flow. This could be, for example, in the supply of the army, ownership of grain barges, or tax farming. Or to give a very concrete example, M. Licinius Crassus made his fortune selling the estates of wealthy senators that were confiscated in Sulla's proscriptions. The elite, although the basis of their wealth was agriculture, would become the very upper tip top Imperial elite through their relation to the state.

In this regards then, no, the wealth of the upper elite was through their position as the "ruling class", even if that meant something very different than it would in Medieval Europe. That being said, one you broader your view outside of this group there seems to have been many more ways to make large sums of money. There are estates in Spain that produced olive oil in massive quantities, enormous sums of money spent of trading voyages in the Indian Ocean, and my favorite, a massive tomb owned by a baker in Rome. Whether these operated through market forces, or whether this wealthy class owned their market situation to personal relations, and if the latter whether we can follow the chain of these associations to the Imperial elite, is heavily debated. I personally prefer a "market economy" model, but there are plenty who see status relationships as being the decisive factor, and even many who agree with a market model like to see this market as inherently conditioned by the imperial system.

Which all doesn't exactly answer your question, but hopefully gets at what you were looking for.

A set of different approaches to this can be found in works like Peter Bang's Roman Bazaar (who articulates an "Imperial" economy), Petern Temin's The Roman Market Economy, and Emanuel Meyer's Ancient Middle Classes but this touches on some truly massive topics.