The Whitney graph 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: 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.
(A late thanks to Asaf!)