Python 2, 83 bytes

n=input() e=m=0 f=2 while~-n: if n%f<1:e+=1;m=max(m,e);n/=f else:f+=1;e=0 print m 

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This answer doesn't do prime checks. Instead, it takes advantage of the fact that the highest exponent of a prime factor will be greater than or equal to the exponent of any other factor in any factorization of a number.