A factorization game Input
A single integer \$1 \leq x \leq 10^{15}\$.
Output
The maximum number of distinct positive integers that have the product \$x\$.
Examples
Input: 1099511627776. Output: 9.  One possible optimal list of factors is: (1, 2, 4, 8, 16, 32, 64, 128, 4096).
Input: 127381. Output 4. One possible optimal list of factors is: (1, 17, 59, 127).
Related to this old question.
Wolfram Language (Mathematica), 52 bytes

Max[Length/@Cases[Subsets@Divisors@#,{a__}/;1a==#]]&amp;


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4-bytes saved thanks to @attinat

Here is also a 153 bytes version that calculates 1099511627776 and 10^15     

Max[Length/@Table[s=RandomSample@Flatten[Table@@@FactorInteger[#]];Last@Select[Times@@@TakeList[s,#]&amp;/@IntegerPartitions@Length@s,DuplicateFreeQ],5!]]+1&amp;      


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The result for 10^15 is 12     


  {1, 2, 4, 5, 10, 16, 25, 40, 50, 100, 125, 250}