The classification of the finite simple groups implies that there are only finitely many finite simple groups of a given exponent $k$. To see this, first note that we can ignore the sporadic groups, as well as the cyclic groups of prime order. It is also also clear that there are only finitely many alternating groups of a given exponent. So we need only consider groups of (possibly twisted) Lie type over finite fields. Here we see that there are only finitely possibilities for the Lie type: otherwise, the Weyl groups would involve arbitrarily large alternating groups. Once the Lie type is fixed, there are only finitely many possibilities for the finite field: otherwise we would obtain semisimple/diagonal elements of arbitrarily large exponent.
The classification of the finite simple groups implies that there are only finitely many finite simple groups of a given exponent $k$. To see this, first note that we can ignore the sporadic groups, as well as the cyclic groups of prime order. It is also also clear that there are only finitely many alternating groups of a given exponent. So we need only consider groups of (possibly twisted) Lie type over finite fields. Here we see that there are only finitely possibilities for the Lie type: otherwise, the Weyl groups would involve arbitrarily large alternating groups. Once the Lie type is fixed, there are only finitely many possibilities for the finite field: otherwise we would obtain semisimple/diagonal elements of arbitrarily large exponent.