The largest permutation group without 2-cycles is $A_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should be polynomially smaller (eg. of size $n!/n^3$), or more dramatically smaller (eg. of size $(.5n)!$).

The largest group I could come up with is {$\phi(x_1,\dots,x_{n/2}) \circ \phi(x_{n+1},\dots,x_n) | \phi \in S_{n/2}$}, which has size $(.5n)!$.