I just saw a problem asking for an example of an algebra of real functions on the interval $[-1,1]$, which do not contain non-zero polynomials and nonzero trigo functions.
I think I just caught one : all the rational functions $\frac{p(x)}{q(x)}$ such that $deg(p) < deg(q)$ and $q(x) \neq 0$ on the interval $[-1,1]$.
However, this one seems a bit weird, do you have any other examples ? Maybe an algebra of functions containing some exponentials, or some logs, or just functions which are not $C^{\infty}$... Any other example appreciated !