Well, we know that a function is Riemann integrable if and only if its set of discontinuities has measure zero. So, you're going to have to have a function that has loads of discontinuities in a very small space. I don't know of any other examples other than the characteristic on the rationals (or irrationals, it doesn't matter which). I'm sure you could construct many such functions, but I think they would all have to be a similarly pathological function to the characteristic on the rationals.

being Riemann integrable does not imply that is Riemann integrable. Counter example:

Let be defined by

Then, clearly, suffers from the same problem that the characteristic function does: its set of discontinuities has measure 1. However, for all . That function is obviously Riemann integrable.