I am only looking for an approximation.
I'm guessing the answer must be somewhere between $10^{20}$ and $10^{50}$.
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Edit:
Okay so my first initial estimation was pretty poor...
I should have realized that $10^{50}$ was equal to the number of atoms composing the entire planet... And even $10^{30}$ was equal to the number of atoms composing the Eiffel Tower, still way too much. Pew's comment gives us an upper bound of about $10^{17}$.
And I've just made a small program that calculated the prime factorization of the first $10^7$ integers in an hour. And I'm pretty sure that more experienced programmers with more powerful computers should be able to compute the prime factorization of at least the the first $10^{10}$ integers.
So we have narrowed the possibilities. The answer must be between $10^{10}$ and $10^{17}$.