Here is a list of other systems:
These system are far older than the current system. How did it get to be known and used internationally by nearly every cultures these days?
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For some history of the Hindu-Arabic system, see e.g. this article and this article. This system was introduced to Europe in the Middle Ages: one of the early influential texts was "Liber abaci" by Fibonacci. See also Wikipedia.
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First of all, the Egyptian and Aegean systems are "agglutinative" rather than positional. I'm looking at the Unicode Aegean numbers block and I see that they have a specific symbol for ninety thousand. What if you need to represent one hundred thousand? Do you just invent a new symbol for it?
Scientists estimate the universe has $10^{80}$ particles. Wolfram Alpha tells me the next prime number after $10^{80}$ is $10^{80} + 129$. If we had some way to indicate exponents in the Aegean number system, we would be able to express both these numbers with that system. But both the Aegean and Egyptian systems are inherently handicapped.
The Babylonian numerals don't have that limitation, but the choice of base is a little unwieldy. And the Europeans didn't even know about the Mayans until after Columbus thought he had discovered a shortcut to India, so the practicality of their system (or lack thereof) was moot.
That leaves the Chinese numerals, and I think the Europeans knew about the Chinese before they knew about the Mayans. But could they use a system that was so intrinsically tied to the Chinese language? Even today I would doubt that.
So as a result of circumstance, we have the Hindu-Arabic numeral system as our primary system of numeration. But do give credit to the Romans: we still use their numerals for some things even today.
For short, positional numeral systems offer the great advantage to have efficient algorithms for the computation of sum and products, easy to use in everyday life. The base $10$ is more or less accidental (besides we having $10$ fingers, on average): for instance, there would be many efficient divisibility tests in base $60$ (since $60$ has a lot of divisors), but $60$ figures are hard to memorize, while in base $2$ "everyday numbers" tend to have too long representations. Base $10$ is a good compromise, even if base $12$ would have probably been better.