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.

Does digits show the first 4000 digits or the last 4000 digits? I would call digits the first 400 4000 digits, but maybe you meant something else.

our minds are only designed to take in so much info. There is a point when we get overwhelmed/overloaded and we shutdown/dont want to respond to anything and I am guessing the same goes for calculators and computer programs. Our brains cant comprehend such large numbers such as my expression 96717311574016^8^16; it has so many digits I have read up that just memorizing those digits, our matter inside our heads may collapse into a "black hole". It has a few quintillion digits and just reading thru all the digits will take us so long I am talking about at least MILLIONS of YEARS, if not MORE! Our brains can only understand a max of maybe a few hundred digits at the most. There is an article that I read up on that stated that our earth is estimated to be only 10^50 atoms wide. Is that the reason why numbers overflow after a certain point?

ok thanks mike! so computing logarithms doesn't work on every single number? I know how to compute a numbers last digits, the first digits are what is extremely hard for me. how do I compute the first 4000 digits of my expression 96717311574016^(8^36 * 14) * 5764801 * 7^3