Overflow while computing 10^16 digit

Questions
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4

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: [WSS20] Implementation of level-index arithmetic for very large numbers

Not sure the status of that project.