Op-amp - Slew-Rate vs Exponential Output
Consider, when the output is slew-rate limited, the response isn't exponential, it's linear. So if there is an exponential response behavior, it has nothing to do with slew rate limiting and removing the slew rate limitation won't remove it.
Now if you study linear systems in general (such as simple RC circuits) you will see that single-pole linear systems have exponential responses (if I understand what you mean by &quot;exponential response&quot;). That is, the response to a step input has the form \$y(t) = A\left(1-\exp(t/\tau)\right)\$
So if you model your op-amp as a single-pole linear system it will have an exponential response. Of course the real op-amp's behavior is only approximately exponential, with the approximation being more accurate the more the op-amp acts like a single-pole linear system.