I once thought of the following variant of your beetle problem. Suppose a kid has a piggy bank with a dollar in it and gets an allowance of a dollar a week (so, this story is implausible). Then, of course, his savings grow without bound. If he has $n$ dollars, then the next week he receives a fraction $1/n$ of his savings, and so $1 + 1/2 + 1/3 + \dots$ must also grow without bound. Tell this to your calculus students but don't analyze it -- it requires infinite products!