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?

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.

For the example number you can use logarithms:

(mantissa = 10^FractionalPart@
     N[(8^36*14) Log10[96717311574016] + Log10[5764801] + Log10[7^3],
      {Infinity, 4001}]) // Precision
(*  4000.64  *)

digits = First@RealDigits[mantissa];
Length@digits
(*  4000  *)

If you want to be more certain, increase the accuracy goal, N[..., {Infinity, 4002}] and compute digits = First@RealDigits[mantissa, 10, 4000].

Of course, Log10[x] does not work on an arbitrary number, only ones for which Log10[x] can be represented a1 Log10[x1] + a2 Log10[x2] + ... in Mathematica.

You could try SLI arithmetic. Someone made a stab at it: https://community.wolfram.com/groups/-/m/t/2030201

Not sure the status of that project.

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.