As suggested in the comments, what you are asking for is essentially the presentation complex of a finitely presented, infinite, simple group. Thus it suffices to exhibit a presentation for such a group. Some are known, but not many.
Probably the easiest examples are Thompson's groups $T$ and $V$. Google gives me a link to an explicit finite presentation for $T$ in §11 of some notes of Levine, based on the classic notes of Cannon, Floyd and Parry.
Even more remarkable examples were constructed by Burger and Mozes. Their examples are CAT(0) amalgams of free groups, and in particular their presentation complex is aspherical. This survey of Caprace is a good place to start learning about these. It looks like the smallest known example is an amalgam of free groups of the form $F_7*_{F_{49}}F_7$ (where the subscripts indicate the ranks of the free groups).
Finally, if you would be satisfied with an almost simple group, meaning a group without non-trivial finite quotients, then Higman's group
$\langle a,b,c,d\mid bab^{-1}a^{-2}, cbc^{-2}b^{-2}, dcd^{-1}c^{-2},ada^{-1}d^{-2} \rangle$
provides a fairly digestible example.