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Wealth Gap Question

Maybe one of our economists can answer this question:

It seems that money begets more money. By this I mean that a rich person has more opportunities to earn a dollar than a poor person does. Many opportunities require a minimum investment, a rich person has more money that they can afford to gamble with, a wealthy person probably better business connections and so on.

This principle implies that the gap between rich and poor should have been growing exponentially since the invention of currency and trade. Whilst the gap is wide, it is clearly checked by something. So, what is the limiting factor limiting the wealth gap? Why isn't the gap even bigger than it is?