At $n=23$, all binomial coefficients are squarefree. Is this the largest value for $n$ where this is the case?
Edit
A plot up to $n=50$:
A plot up to $n=500$:
plotted against $n+1$ and $\frac{112}{\sqrt{239}}\sqrt{n}$
and the same up to $n=2000$:
Do these bounds hold for all $n$?
(Clearly, the bound $n+1$ holds for all $n$.)
Update
Just out of interest, a plot up to $n=2000$ with bounding curve $24\log(n)$
($24\approx\frac{148}{\log 479}$, where $148$ is the number of squarefree binomial coefficients at $n=479$),
which seems to be the tightest curve which still holds for up to $n=3967$:
... seems to suggest that the bound is $const. \log(n)$, which so far shows $c\approx24$ (which, coincidentally, is the number of squarefree coefficients at $n=23$).
Filtered([1..100000], n -> ForAll([0..n], k -> MoebiusMu(Binomial(n,k))<>0));
$\endgroup$ – Jean-Claude Arbaut Nov 28 '13 at 16:41