I am just guessing here, but ...

If we were dealing with real numbers, I believe that when breaking a number 'x' into equal pieces which sum to x, and which have a maximal product, you should divide x by e to get the number of pieces.  Each piece will be e.  

I suspect that with integers, the answer will be similar.

2024/e is between 774 and 775, so I suspect the maximal product for the integer 2024 will be with 774 or 775 pieces, each being either 2 or 3.

For example 208 2s and 536 3s.  Or 211 2s and 534 3s.

So the product could be something like 2^211 * 3^534.

Also, the larger x is, the larger the final result.

So I expect the largest product of the century will be with 2099, for example    2^217 * 3^555

These are just guesses.