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 

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> (***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).
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(A late thanks to Asaf!)

Answer: 
Not sure if this fits the scope of the question, it's hardly as advanced as all the other suggestions but I find it quite elegant and surprising:

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> There exists no positive integer sandwiched between a perfect square
>  and a perfect cube, with the sole exception of $26$ - equivalent to saying that the only solution of $a^2 \pm 2 = b^3$ is $(5, 3)$.
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Of course this is just one among many Diophantine equations which admit only one solution, but it has a special significance when viewed from a less abstract point of view.

Another nice one:

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> For any prime $p$, the product of its primitive roots is congruent to $1$ modulo $p$, except for $p = 3$ for which the product is equal to $2$ (result due to Gauss).
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Every family of graphs that is closed under minors has at least one forbidden minor, with the exception of the family of all graphs.