Warning: not an answer. Rather, some comments and some links.

I came upon this issue myself when I was teaching an undergraduate real analysis course some years ago. The point is that in the development of the Riemann/Darboux integral, a standard technical result is that if $f: [a,b] \rightarrow [c,d]$ is integrable and $\varphi: [c,d] \rightarrow \mathbb{R}$ is continuous, then $\varphi \circ f$ is integrable. It follows easily that the product of two integrable functions is integrable (which is not so obvious otherwise). This result appears, for instance, as Theorem 6.11 in Rudin's *Principles of Mathematical Analysis*.

It is easy to see that the composition of integrable functions need not be integrable. So it is natural to ask whether it works the other way around. Remarkably, I know of no standard text which addresses this question. Rudin immediately asks a much more ambitious question and then moves right on to something else.

At the time I convinced myself of the existence of continuous $f$ and integrable $\varphi$
such that $\varphi \circ f$ was not integrable. However, in order to do so I needed to use ideas which were more advanced than I could explain in my course. At least, this is what it says on p. 7 of my lecture notes:

http://math.uga.edu/~pete/243integrals2.pdf

Unfortunately I didn't write down the counterexample that I had in mind (I suppose back then I was clinging naively to the idea that the lecture notes were for the students and not to preserve my own knowledge of the material in the coming years), so I don't know now what it was.

The example in Jitan Lu's 1999 Monthly article that Qiaochu referred to seems elementary enough so that it should at least be referenced in texts and courses, and possibly included explicitly. For those who couldn't get the whole paper from the previous link, it is now also available here:

http://math.uga.edu/~pete/Lu99.pdf

Of course, I don't believe for a second that an example of this type (i.e., to show that integrable $\circ$ continuous need not be integrable) was first constructed in 1999. Can anyone supply an earlier reference? (I never know how to go about solving math history problems like this.) I should say that I am impressed that Qiaochu was even able to track down this paper. The MathSciNet review is quite unhelpful. It says:

In this note the following result is given. If $f$ is a Riemann integrable function defined on $[a,b],\ g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $f\circ g$ is Riemann integrable on $[c,d]$.

This is not the main result given in the paper; rather it is a proposition stated (without proof!) at the very end.