exotic ℝ⁴
Euclidean space R^n is the blandest manifold imaginable, and for every dimension except one there is exactly one way to put a smooth (calculus-compatible) structure on it: any two smooth structures on R^n are diffeomorphic. The exception is n = 4, and it is a spectacular one. There exist smooth manifolds that are homeomorphic to ordinary R^4 — topologically indistinguishable from it — yet are NOT diffeomorphic to it: you cannot smoothly identify them with standard R^4 no matter how you try. These are exotic R^4's, and there are uncountably many distinct ones. Dimension four is the only dimension where R^n has more than one smooth structure, and it has continuum-many.
How can this be? The result is forced by combining two opposite-direction theorems. Freedman's work (1982) on topological four-manifolds classifies them by algebraic data (the intersection form plus the Kirby-Siebenmann invariant) and, with surgery, produces topological R^4's. Donaldson's gauge theory (1983) constrains which smooth four-manifolds can exist (the diagonalizability of definite intersection forms). The clash between what topology allows and what smoothness forbids produces, almost as a byproduct, a smooth open four-manifold that is homeomorphic but not diffeomorphic to R^4. The first examples came from Donaldson plus Freedman; Taubes then showed there is a continuum (an uncountable family) of pairwise non-diffeomorphic exotic R^4's, distinguished by gauge-theoretic invariants. A 'small' exotic R^4 embeds smoothly in standard R^4; a 'large' one does not — there are infinitely many of each kind.
Why it matters and what to be honest about. Exotic R^4 is the cleanest possible demonstration that smooth structure is genuinely more than topology, and that dimension four is exceptional — high dimensions are tamed by the h-cobordism theorem and surgery, dimensions two and three are too rigid to allow exotica on Euclidean space, and only four sits in the gap where the techniques of neither low nor high dimensions apply. Crucial cautions. First, 'manifold' never implies a canonical smooth structure: R^4 admits many, so writing 'R^4' is ambiguous about which smoothing you mean unless you say standard. Second, this is special to NON-compact R^4; the analogous open question for, say, exotic structures on closed four-manifolds (and the smooth four-dimensional Poincare conjecture, whether S^4 has an exotic structure) is famously still open — do not conflate 'exotic R^4 exists' with 'every four-manifold has exotic structures' or with any statement about S^4. Third, no exotic R^4 has ever been written down by explicit equations; their existence is proved by the contradiction between Freedman and Donaldson, which is part of why they remain so mysterious.
The standard R^4 contains the standard radius-1 four-ball smoothly. A 'small' exotic R^4 is a smooth manifold homeomorphic to R^4 that contains a compact set K which no smoothly embedded standard S^3 can surround inside the exotic structure — the gauge-theoretic obstruction (a Donaldson-type argument on a related closed manifold) shows such a smoothing exists and differs from standard R^4.
A compact set that cannot be enclosed by a smooth standard S^3 distinguishes a small exotic R^4 from the standard one.
R^n has a unique smooth structure for every n except n = 4, where there are uncountably many; this is the single most striking fact pinning down how special dimension four is, and it should never be stated as if some other R^n were also exotic.