Is it possible to have a rate of change of acceleration?

Your question is not weird; it is legitimate. It is possible, it exists, can be of use and it is called jerk, jolt, surge or lurch, and is defined by any of the following equivalent expressions: $$\vec j(t)=\frac {d\vec a(t)} {dt}=\dot {\vec a}(t)=\frac {{d}^2 \vec v(t)} {dt^2}=\ddot{\vec v}(t)=\frac {{d}^3 \vec r(t)} {dt^3}=\overset{...}{\vec r}(t)$$

It is useful in the Dirac-Lorenz equation (as Emilio linked).

In case you are asking yourself, a fourth derivative (rate of jerk) is also defined, and it is called jounce

This does exist, and it is called Jerk, see the Wikipedia page on jerk.

It is used quite frequently in physics concerning humans, as we are able to sense this, and there are limits to how much jerk a human can endure.

It is, however, quite abstract and therefore more difficult to comprehend, which might be the reason that lower level textbooks do not mention it.

There are many reasons why acceleration would not be constant. Books often don't mention it because they are getting mathematicians used to the concept of first derivatives before moving onto second, third etc...

Consider the following example which ends up leading to Tsiolkocsky's Rocket Equation: http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

A rocket has mass, and propellant. To accelerate the rocket expends propellant over time at a fixed rate, giving a constant force. This means the Mass changes over time at a fixed rate, and if you look at f=ma, you quickly realize that acceleration is not constant, as acceleration becomes a function of a variable mass.

In this case total mass is a second derivative of delta-v (total change in velocity).