What lies beyond the Sedenions In the construction of types of numbers, we have the following sequence:
$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$
or:
$$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$
or:
"Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"
With the following "properties":


*

*From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering).

*From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity.

*From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity.

*From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.


The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?
 A: I believe John Baez has answered your question in a series of short articles/blogposts which you could begin for example here.
A: What you are talking about is precisely the Cayley-Dickson construction. 
Remark: I am left wondering what is gained by going past Octonions.  The the first 4 are very special as they are the unique 4 normed divison algebras over $\mathbb{R}$.  Perhaps someone with more knowledge can point out the possible uses of the Sedenions and their higher counterparts.
A: If with "beyond" you mean something like "after" the answer is yes: the Cayley-Dickson's construction has no end and you can always extend your number system to a higher dimension one (always with $2^n$ dimensions with $n\in \Bbb N$), after Sedenions $\Bbb S$ come Trigintaduonions $\Bbb T$ which someone say are useful in electrical and computer engineering (I can't really figure out how) but they don't loose any other properties more than Sedenions do, this set and the others that came after it are not very interesting (with an only mathematical point of view) cause of that, they just have some subgroups that look like the Sedenions.
If with "beyond" you mean something like "other than" the answer is still yes: there are a lot of hypercomplex numbers with 16 dimensions system like the extension of tessarines, duals and hyperbolic number and so on.
A: You can build tessarines of any dimension of the for $2^n$. They still will be commutative ans associative. This is used in digital signal processing.
Cayley-Dickson's construction, used in quaternions, on the other hand, loses most of its useful properties with the growth of the number of dimensions.
