If Colmin(N) is the lowest value reached by iterating the process from initial value N, Tao proved that

Colmin(N) < f(N)

for arbitrary f provided f tends to infinity, for almost all N. Here, "almost all" means something like "exceptions(N) / N, tends to 0 for large N", where exceptions(N) is the count of values that do not obey the inequality [0]

So it's an asymptotic result. The first million integers could all be exceptions - but eventually the proportion of exceptions dies out.

The ability to pick arbitrary f is very powerful. Pick the slowest-growing function you can think of. e.g. Tenfold-iterated logarithm. The inequality says for all but a negligible fraction of integers, Colmin grows slower than that function.

[0] Nitpick: I'm describing the natural density, but Tao needs the logarithmic density, where each exception n is weighted by 1/n.