According to this webpage and this mathworld article, if $G<S_n$ is a permutation group which acts sextuply transitively then $G=A_n$ is the alternating group, but this fact is known on the basis of the classification of finite simple groups. Is there any $k$ for which we know that $k$-transitive implies alternating without assumption of the enormous theorem, or any other advanced technology?
One could also ask about $(n-r)$-transitive groups of degree $n$ for fixed $r$. In particular, $r=1$ yields only symmetric groups, and a counting argument with $r=2$ yields only symmetric and alternating groups. The next stage would be to ask, can we prove $(n-3)$-transitive groups of degree $n$ are symmetric or alternating with only finitely many exceptions without invoking the enormous theorem?