I'm no expert but I did study Marx, history and comparative political science at university. First off there has never been a truly communist country as Marx defined it. He lists stages of development towards communism, which itself is really a benign for of anarchy. Basically it goes: Revolution, the dictatorship of the proletariat, then eventually communism. The classic example of the USSR is the dictatorship phase. Nobody ever got past that.

Now in terms of wealth the idea was the everyone would be paid the same wage, regardless of your job. However it was more complicated then that. There was a sort of class system where the upper classes such as party members, important skilled people (doctors, scientists, etc), and high ranking military officers enjoyed perks that a factory worker did not. At the elite level people lived like kings compared to most other citizens. Even though they notionally were paid the same, all of these "extras" were off the books. However, even for these elites, they were still stuck in the closed off economy of the USSR; meaning that there was little travel, consumer goods, or entertainment options to enjoy compared to the west. It would have been much better to be rich in the West then rich in the USSR.

I was wondering: Was it possible for an individual living in a purely communist country like the Soviet Union, Cuba or other Eastern Bloc countries to become exceptionally rich compared to the rest of the population?

There have been questions of that sort before. Problem with wealth in the Soviet Economy was you couldn't buy a lot of stuff with money. You couldn't own more than one flat and one dacha. You couldn't have several cars. When there was a shortage on, say, toilet paper, you received a set amount per head regardless of how much you were willing to pay.

Even if you had a lot of money you still had to work since unemployment was a crime. Your holiday destinations, your ability to have your daughter attend an important concert, to study in a prestigious university or to travel to prestigious holiday destinations was a matter of goodwill from bureaucrats independent of your bank account. And the bureaucrats have mostly shown goodwill to other influential people, like the other bureaucrats, to teachers, or, say, to medics treating their relatives. (See Alena Lebedeva's "Russia's Economy of Favours: Blat, Networking and Informal Exchange". )

However: Khrushchev introduced some luxury goods (cars, cameras, fur coats) to work as a money sink (to make f.e. higher wages incentives to go to Northern Siberia work). Aside from that there were "Beryozka" stores for foreigners only and there were special luxery goods provided to, f.e., the scientists willing to live and to work in Akademgorodok or, say, to officer families living in similarly hazard regions. (see "An Economic History of the USSR from 1945" by Philip Hanson)

This reply may be below the standarts of this sub, since the book I'm about to mention is a fictional one, albeit immensely influencial in the Russian culture. You may want to read a satirical novel The Golden Calf (1931) by Ilf and Petrov. From the summary:

Ostap Bender hears a story about a "clandestine millionaire" named Alexandr Koreiko, who has made millions through various illegal enterprises by taking advantage of the widespread corruption in the New Economic Policy (NEP) period while pretending to live on an office clerk's salary of 46 rubles a month. Koreiko keeps his large stash of ill-gotten money in a suitcase, waiting for the fall of the Soviet government, so that he can make use of it. Bender finds out about Koreiko and starts to collect all the information he can get on his business activities. <...>
Suddenly rich, Bender faces the problem of how to spend his money in a Communist country where there are no legal millionaires. Nothing of the life of the rich that Bender dreamt of seems possible in the Soviet Union.