A full classification of subgroups of $GL_n(q)$ containing primitive prime divisors of $q^n-1$ is available in the literature. See here:

Robert Guralnick, Tim Penttila, Cheryl E. Praeger, and Jan Saxl. Linear groups with orders having certain large prime divisors. Proc. London Math. Soc. (3), 78(1):167–214, 1999.

One needs only find which of the groups mentioned in this paper lie in $SL_n(q)$ to answer the OP's question.

In addition to the examples that Derek mentions there are also, in general, field-extension subgroups (i.e. Aschbacher's $\mathcal{C}_3$ class) whenever $n$ is a composite. So, for instance $SL_6(q)$ contains $SL_3(q^2)$ and $SL_2(q^3)$, both of which are non-abelian and have primitive prime divisors of $q^6-1$ for $q>2$.

And, as Derek mentions, there are also loads of sporadic examples.

In fact the paper cited above doesn't just deal with ppd's of $q^n-1$ but ppd's of $q^e-1$ for $e>\frac{n}{2}$.

A full classification of subgroups of $GL_n(q)$ containing elements of order a primitive prime divisor of $q^n-1$ is available in the literature. See here:

Robert Guralnick, Tim Penttila, Cheryl E. Praeger, and Jan Saxl. Linear groups with orders having certain large prime divisors. Proc. London Math. Soc. (3), 78(1):167–214, 1999.

To give a full list of counter-examples to the OP's question, then, one needs only find which of the groups mentioned in this paper are non-abelian and lie in $SL_n(q)$.

In addition to the classical groups that Derek mentions the cited paper lists a number of `geometric' subgroups that satisfy the required conditions. These include the field-extension subgroups (i.e. Aschbacher's $\mathcal{C}_3$ class) whenever $n$ is a composite. So, for instance $SL_6(q)$ contains $SL_3(q^2)$ and $SL_2(q^3)$, both of which are non-abelian and have primitive prime divisors of $q^6-1$ for $q>2$.

Apart from these, all counter-examples are `nearly simple' i.e. their projective image is an almost simple group. These include the sporadic examples mentioned by Derek plus a bunch of others.

In fact the paper cited above doesn't just deal with ppd's of $q^n-1$ but ppd's of $q^e-1$ for $e>\frac{n}{2}$.