Question: The [Whitney graph isomorphism theorem](https://en.wikipedia.org/wiki/Line_graph#Whitney_isomorphism_theorem) gives an example of an extraordinary exception: a very general statement holds except for **one** very specific case.

Another example is the [classification theorem for finite simple groups](http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Statement_of_the_classification_theorem): a very general statement holds except for **very few (26)** sporadic cases. 

I am looking for more of **this kind** of theorems-with-not-so-many-sporadic-exceptions 

> 
> (***added:***) where the exceptions don't come in a row and/or
>  in the beginning - but are scattered truly [sporadically](https://en.wiktionary.org/wiki/sporadically).
> 
> 
> 

(A late thanks to Asaf!)

Answer: 
For squarefree positive integer $d$, every imaginary quadratic field $\mathbb{Q}(\sqrt{−d})$ has class number greater than 1 unless $d$ is equal to one of the Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163.
No free group is amenable, with the exception of $\mathbb Z$.