People have been shocked precisely that breeding rabbits grow exponentially. In particular the Australians were in the 19th century. In 1859, some guy named Thomas Austin brought 24 rabbits from England so he could amuse himself by shooting at them. In a few years the population had grown to hundreds of millions of rabbits, decimating Australia's native ecology. Oops. (link: http://en.wikipedia.org/wiki/Rabbits_in_Australia)
Another simple example is money. You invest $1,000 in a bank account earning 10% compounded continuously; how much do you have after 30 years? The answer: A whole heck of a lot of money. I think it is not all that intuitive to many people, but the calculus unambiguously proves it.
I have also covered predator-prey models (which are covered in Stewart's Calc I book.) The differential equations are simple to explain, but draw the oval in the phase plane, and to a beginner the behavior is really rather stunning: the populations oscillate! I think the most natural naive guess is that the populations' behavior would settle down to some equilibrium, and this is not what happens. Probably you can even compare the behavior of the ODE's with data from some real world example (I think Stewart has a graph of such data).
Perhaps you're teaching some kind of honors class where this is old hat to the students; I confess that I'm somewhat disagreeing with the premise of your question, so perhaps I've misunderstood the situation you're in. But in my experience, there is a lot of meat in the very simplest examples.