If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The Frobenius theorem says that the only finite-dimensional associative division algebras over $\mathbb R$ are exactly $\mathbb R$, $\mathbb C$ and $\mathbb H$, up to isomorphism.

The octonions are not on this list, but they are not very well-behaved either; their multiplication is not even associative. And it only gets worse as you move further into the Cayley-Dickson sequence.

(As Fabio Lucchini points out, Hurwitz's theorem states that if you don't require associativity, but still want the algebra to have inverses and a norm that agrees with the multiplication, you get $\mathbb R$, $\mathbb C$, $\mathbb H$, and also $\mathbb O$, but nothing more).

At some point things get so bad that one may wonder if you would not rather consider, for example, $\mathbb R^3$ with the cross product. Or just declare any random bilinear map $V\times V\to V$ to be your multiplication.

Henning Makholm's answer explains this well. But here's another way to think of it, if you want some intuition on why it works that way.

The real numbers, complex numbers, and quaternions are all related to geometric algebra. In particular, think about how complex numbers represent rotations and scaling in two dimensions: $i$ is a ninety-degree rotation in the plane. Similarly, quaternions represent rotations and scaling in three dimensions, with $i, j, k$ acting as ninety-degree rotations in three orthogonal planes ($xy, yz, zx$). And if you extend this, you can think of the real numbers as representing "rotations and scaling" in one dimension, where there's no plane to rotate in, so it's really just scaling.

What happens when you extend this to four dimensions and above? Well, rotations in four dimensions get complicated. In three dimensions and below, every rotation is "simple": it can be represented as a plane of rotation and an angle within that plane. In four dimensions, that stops working. Because in four dimensions, you can have two planes that aren't parallel and also don't intersect (such as the $xy$-plane and the $zw$-plane). If you rotate in both of those planes at the same time, there's no single "plane of rotation" any more.

The rotations-and-scaling representation in four dimensions would have eight elements $\langle 1, xy, xz, xw, yz, yw, zw, xyzw\rangle$, where the last one is what you get when you multiply $xy$ by $zw$ and can't really be visualized except as a "directed hyperspace". (Formally, it's a quadvector, aka a four-blade, but neither of those words really helps.) And because rotations are no longer simple, the math stops working quite as nicely, so this one doesn't get its own name: it's related to the octonions, but not quite the same.

You can keep going further, but the math just gets messier and messier and less and less elegant. If you try this in five dimensions, you get a relative of the "sedenions", with sixteen elements. And so on and so forth. The sedenions are useful if you want to work with rotations in five dimensions, but we don't generally need to do that, and they're really ugly and not that exciting in general. So most people just don't care about them.

If you want to look into this further, the "scalings and rotations" are formally called the even Clifford sub-algebra in $n$ dimensions, written as $Cl^{+}_{n}$ or $Cl^{[0]}_{[0,n]}(\mathbb{R})$. (It sounds a whole lot more complicated than it is.)

TL;DR: the reals, complexes, and quaternions have the number of elements they do because they correspond to rotations and scalings in a particular space. Other numbers of elements don't have that correspondence, and higher-dimensional spaces get messy and become less elegant.