I often read that Crassus was not only the richest person of his day and throughout Rome, but possibly in history. While listening to The History of Rome podcast, Mike Duncan said that the Emperors personally accounted for about half of the GDP of the entire empire. Wouldn't they be wealthier than Crassus ever was?
Crassus' wealth was overblown even in his own day. It's possible that early in Crassus' career he might have made the claim that he was the wealthiest man in Rome stick, but by the time Pompey returned from his campaigns in Asia Crassus was most certainly not the wealthiest man in Rome. Crassus' wealth was based mainly in property and slaves which he ate up during Sulla's proscriptions, buying or seizing the property of the proscribed. Pliny estimates Crassus' wealth at 200 million sesterces, which is a chunk of change. Thing is, Crassus got his reputation mainly through his ruthlessness in obtaining his wealth and his showiness when displaying it. In reality, when Pompey returned from his campaigns the income he got from land and tribute among his subjects and client-states in the east far outvalued anything that Crassus could even dream of. Just to put this into context, the city of Rhodes yielded to Cassius 200 million sesterces, the value that Pliny says was Crassus' entire fortune. Now consider that Pompey had control of the tribute and lands of every city from Bithynia to Judaea, most of which approached Rhodes' wealth and many of which surpassed it. Crassus' fortune was chump change to Pompey. Even Caesar probably outvalued Crassus by the time of Crassus' death, since the slaves and tribute that he could exact from his territories in Gaul probably outdid the value of Crassus' land. In fact, Crassus' campaign in Parthia was probably less motivated by a desire to match the glory that Caesar and Pompey had gained in their campaigns (it was getting a bit late in the game for that) but by a desire to increase his wealth to match their own by gaining tribute before he fell permanently behind.
There was little distinction between the Emperor's finances and the Empire's finances. Crassus might've been one of the wealthiest private citizens, but the Emperor had access to all of the tariffs, taxes, and tithes in the whole Empire. So, the Emperors were infact wealthier. According to Pliny, Crassus had a networth of 200 sesterces. In contrast, the Emperor Augustus spent 700 million sesterces just to buy land to settle soldiers on. That's just counting a chunk of the money he spent, and not how much he even had.
Sources: Anthony Everitt, Augustus and Walter Eder, The Cambridge Companion to the Age of Augustus