What are some function spaces that are UFDs? What are some "useful" sets of functions (rings under pointwise multiplication) $\mathbb{R} \rightarrow \mathbb{R}$ that are unique factorization domains, other than the polynomial ring?
I will quantify usefulness in my context: I'm only interested in functions that form an algebra over $\mathbb{R}$ (or $\mathbb{C}$) in the space of continuous functions, and preferably also forms a dense algebra in $C([a,b])$ with respect to the usual supremum norm.
 A: This is a very nice question.
I'll boldly  assert that outside of algebraic geometry there are probably no examples of spaces of functions that are unique factorization domains!
The reason is that most spaces of functions, be they continuous, differentiable or smooth (=$\mathcal C^\infty$) do not even constitute a domain.
Holomorphic functions on a connected open subset  $U\subset \mathbb C$ (or on a connected holomorphic manifold  of arbitrary dimension) do constitute a domain $\mathcal O(U)$ , but (unless I'm mistaken)  are never unique factorization domains.
This is easy to see for the entire functions $\mathcal O(\mathbb C)$ : irreducible entire functions  are of the form $az+b$ and $\sin(z)$ is certainly not a finite product of such affine functions. 
In algebraic geometry however there do exist affine varieties $X$ for which the regular functions $\mathcal O(X)$ form a unique factorization domain : for example $\mathbb C$ or $\mathbb C^*$.
And affine varieties for which this is not the case, like the quadric in $\mathbb C^3$ given by the equation $z^2-xy=0$.
Deciding which is which for an affine variety is a very delicate problem.
And to finish on a slightly humorous note, let me remark that the domain of trigonometric polynomials $\mathbb R[\cos(\theta), \sin(\theta)]\subset \mathcal C(\mathbb R)$ is not a unique factorization domain because of the reasonably well known formula         $$\cos\theta \cdot \cos\theta=(1-\sin \theta)\cdot (1+\sin \theta) $$
A: The ring of germs of functions holomorphic at the origin?
