My question is simple, but I haven't seen it to be addressed anywhere:
What would be a simple example of an affine variety that is not a toric variety?
Toric varieties (the ones I have studied) are constructed by a fan using "gluing". (For some examples, see Example 2.2 in Page 20). So to prove that some affine variety is not toric amounts to showing that there is no possible fan that gives rise to this variety. And I am not sure how could one exactly do that.
An example of projective variety that is not a toric variety is also welcome.
By the way, some authors require toric varieties to be normal varieties, but some do not. I am not looking for an example of such variety as an answer to this particular question. (So I really want something that's not "toric" in all senses of the word). I am especially interested in knowing how it is possible to prove that a variety cannot be obtained from a fan using gluing.
Complex toric varieties are rational, i.e., birational to projective space. So any affine curve of positive genus is not an affine toric variety.
Added: If $f(x) \in \mathbb{C}[x]$ is a polynomial with distinct roots and of degree at least $3$, then $y^2 = f(x)$ has positive genus. (More precisely, if $f$ has degree either $2g+1$ or $2g+2$, then the genus is $g$.) So maybe the simplest example is $y^2 = x^3-1$.