I seem to recall that Avogadro's number is 6.02 x 10^23.

How exact is this number and how is it derived?

It is often used as 6.022 X 10^23. I think my high school text books used 6.023 X 10^23. That, along with it never being given, suggests we don't know the exact value of the fourth decimal. When I was teaching math, I used it as a good illustration of scientific notation, and not making it look like a number is much more precise than it is.

I don't remember how it was determined.

The current best estimate is (6.02214179 +- 0.00000030) x 10^23

It can be measured using crystallography, as you know how many atoms there should be by measuring the crystal spacings using x-ray crystallography. They usually use silicon.

The fourth decimal certainly is known. Labman, it is never given for the same reason that only a few digits of pi are ever given, not because we don't know it, but because in chemistry a 4 decimal place precision is almost never needed.