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: 
How about the Ax-Kochen theorem?

Every homogeneous polynomial of degree $d$ in $n$ variables with $n>d^2$ has a non-trivial zero in $\mathbb{Q}\_p$ for all but finitely many $p$... and the finite set of exceptions depends on the degree $d$.

Here are some things we know about the finite exception set for various $d$: <http://en.wikipedia.org/wiki/Ax-Kochen_theorem#Exceptional_primes>

This is a special case of the more general fact that any first-order sentence (in the language of valued fields) which is true of all but finitely many Laurent series fields $\mathbb{F}\_p((t))$ is true of all but finitely many $p$-adic fields $\mathbb{Q}\_p$. Model theory gives us many more examples of the principle "true in char $0$" = "true in char $p$ for large enough $p$".
Just about every single theorem in arithmetic that concerns prime numbers fails for $2$.