Which mathematical claims are true except for exactly one case?

The most striking one I know is about ordinary space itself.

For every dimension n, there is exactly one way to do smooth calculus on n-dimensional space — one smooth structure, up to equivalence. That is true for dimension 1, 2, 3, 5, 6, 7 and every dimension above.

Except dimension four, where there are uncountably many.

What makes this the best example in the category is how badly it fails. It is not that dimension four has two, or finitely many extra. It has infinitely many, more than there are whole numbers. Every other dimension is boring in exactly the same way and one is spectacularly, uniquely not.

And of course four is the dimension we appear to live in, which nobody has ever quite been able to leave alone.

My favourite from group theory: for symmetric groups — the groups of all rearrangements of n objects — every symmetry of the group is an inner one, meaning it comes from conjugating by an element of the group itself.

True for every n. Except n = 6, which has an extra symmetry that comes from nowhere and cannot be explained by conjugation.

There is no small reason for it. Six is not special in any way that shows up first; the exception is discovered by working through the proof and finding that one step fails in exactly one place. It has been connected to several other unique objects that also live at six, which suggests something coherent is going on, and it still reads as an accident.

A related one: the finite simple groups fall into a small number of infinite families, plus twenty-six sporadic groups that belong to no family at all. That is not one exception, but it is the same flavour — a clean classification with a bag of leftovers that stubbornly exist.

One with a very short statement: 8 and 9 are the only pair of consecutive perfect powers.

That is, among all numbers that are a whole number raised to a whole power greater than one — 4, 8, 9, 16, 25, 27, 32 and so on — you will never find two that differ by exactly one, other than 8 and 9.

It was conjectured in the middle of the nineteenth century and only proved at the start of this one. So for about a hundred and fifty years everyone was confident there was exactly one exception without being able to rule out a second hiding somewhere enormous.

That gap between "obviously true" and "proved" is its own kind of interesting.

A gentler one for anyone who wants an example they can verify themselves: among the regular polygons, the square is the only one whose perimeter and area are numerically equal for a whole number side length — side 4 gives perimeter 16 and area 16.

That one is engineered enough that it may not meet your non-trivial bar, but it is the kind that works well with students, because they can find it by hand and then be told that the grown-up versions of the same phenomenon go all the way up to the structure of four-dimensional space.