If $s(n)$ denotes the largest integer such that $\binom{n}{k}$ is divisible by $p^{s(n)}$ for some prime $p$ for at least one $1\leq k<n$ then it is easy to see that $s(n)\to \infty$ as $n\to \infty$ (and in fact that $s(n) \asymp \log n$).

This problem was solved in the affirmative by Cambie, Kovač, and Tao (see the comment section). A Lean formalisation of their proof is available here.

There are other simpler constructions: for example, $3^{2^k}$ for arbitrarily large $k$ (see this discussion).

See also [175].

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