According to an email purportedly shared between Nakamoto and Bitcoin Core contributor Mike Hearn, Satoshi reasoned that if 21 million coins were to be used by some fraction of the world economy, 0.001 BTC (1 mBTC) could be worth around €1. This prediction came true back in 2013, when Bitcoin first broke through the €1,000 price point; today, each mBTC is currently worth closer to €8.25.

Although Satoshi compares the price of Bitcoin to the Euro in his email, some simple mathematics indicates he may have had a much grander vision for Bitcoin—better explaining why the 21 million maximum figure was chosen.

At the time of Bitcoin's creation, the entire world's money supply stood at approximately $21 trillion. This figure, known as the M1 money supply, is made up of the total value of all the physical money in the world, including cash, coins, travelers' checks, and more.

If Bitcoin were to grow to become the single world currency—replacing all those that the M1 figure is comprised of—then each BTC would be worth $1 million. Because there are 100 million satoshi in each Bitcoin, this would place the value of each satoshi at $0.01.

Calculate the number of blocks per four year cycle:

6 blocks per hour
* 24 hours per day
* 365 days per year
* 4 years per cycle
= 210,240
~= 210,000

Sum all the block reward sizes:

50 + 25 + 12.5 + 6.25 + 3.125 + … = 100

Multiply the two:

210,000 * 100 = 21 million.

Or just look at the graph:
...
Obviously it won't ever cross (or even reach) 21 million.