The Weierstrass theorem tells us that any continuous function can be approximated arbitrarily closely by polynomials (which is what Bézier curves are). And, in fact, Bernstein's proof of this theorem uses polynomials expressed using the Bernstein basis (i.e. Bézier curves). So, by using a Bézier curve of sufficiently high degree, you can approximate any curve as closely as you like, even without the "no sharp corners" restriction that you mentioned.
This approximation technique (in the form used by Bernstein) isn't really practical because huge degrees are required to get decent accuracy.
You can approximate any curve by a (cubic) spline, too, and a cubic spline is just a sequence of cubic Bézier curves joined end to end. This approximation technique certainly is practical, and is very widely used.
If you want exact representations, rather than approximations, there are many curves that can not be handled. As you mentioned, circles are the simplest ones.