Just to be clear, 8^16 is nowhere near the biggest number Mathematica can compute. I think you meant that the max contains 10^16 digits, like you said in your first sentence. You can ask for what the biggest number Mathematica can handle is with $MaxNumber.
As for why, it's obviously because computers have finite memory. Mathematica can't compute a number that it can't actually represent in the memory available on the computer where it's running.
As for how to compute the 4000 leading digits of your number, I think that's more of a math question. Are there algorithms for computing the leading digits of a product without first computing the trailing digits? Seems infeasible to me, but I'm not a mathematician.
