Wait, is getting really rich the standard? Because not all languages are equal in their ability to ring this bell at any given time. Right now, English is king -- its the lingua franca of half the world or more, and has a disproportionate number of populous rich countries using it as their first language -- America is conspicuously important in this regard.

Writers in other languages are at a severe disadvantage in terms of ability to cash in on their local popularity and influence. I understand that Russia, for instance, takes poetry very, very seriously; but its legal and economic system is not capable of giving its poets their honors in the form of money. For example, Yevgeny Yevtushenko is a big deal, and he's not starving as far as I can tell. But neither does he make anything like a tenth of what cheesy American hack (not hacker -- the bad, journalistic kind of hack) Dan Brown makes.

Speaking of Dan Brown -- he's a huge financial success, but he's the non-thinking man's version of Umberto Eco, who, from a financial point of view, made the extremely unwise decisions to be Italian and not to pander to the lowest common denominator among his potential readers.

Historically poets sometimes made large sums of money:

"One of the first authors to successfully take advantage of the statute was Alexander Pope, who became the equivalent of a millionaire by writing his own verse translation of The Iliad (between 1715 and 1720) and The Odyssey (in 1726). By 1727 Pope made more than £8000, the equivalent of about $12 million today and the largest sum an English author had ever received for their work. But stories of wealthy 18th century authors are rare. Since few people were literate, and fewer still were at leisure to read literature, there was little monetary incentive for professional authors to write it."

http://www.mcsweeneys.net/links/stateofpublishing/authors.html

And less scientifically, some cultures had traditions of court poets being paid large sums - I seem to recall some Arabic poems speaking of the poet's "mouth being stuffed with gold" by the caliph for their ghazals, and Scandinavian skalds could be rewarded with substantial amounts of gold for a good drapa.

Who's wealthy in the present? Well, that's a little harder. In the Anglosphere, I'd wonder what the net wealth of Seamus Heaney and Robert Frost are/were, since they seem to be some of the rare crossover successes. (I'd exclude Wallace Stevens since being an insurance executive probably paid pretty well.)

It's worth noting that patronage is still a viable strategy in some areas; here's one living painter you've never heard of who is estimated to be worth $114 million (and was not born or married into wealth). (Damien Hirst, who you have heard of, is somewhere around $350 million.)