Are there useful algebras between the Cayley-Dickson algebras? Reals, complex numbers, quaternions, octonions, etc are a hierarchy of algebras which can be constructed in a regular way.
One obvious property of this hierarchy is that each such algebra has $2^n$ natural basis elements.
Are there related algebras which have 3, 5, 7, or some other intermediate number of natrual basis elements?
If not, why not?

Well, you can always equip $\mathbb{R}^k$ with the product $(\mathbf{x}\ast \mathbf{y})(i)=\mathbf{x}(i)\cdot \mathbf{y}(i)$ to give you a unital associative commutative algebra, but for all $k\geq 2$ you won't have this be a division algebra.  
The Frobenius Theorem says that the only real division algebras are $\mathbb{R},\mathbb{C},\mathbb{H}$ (up to isomorphism).  
Other standard algebras include 


*

*group algebras (if $G$ is a group, consider the free $\mathbb{R}$-vector space generated by $G$, where we define the multiplication by the group law and extend linearly) 

*Clifford algebras which actually produces the examples of $\mathbb{R},\mathbb{C}$, and $\mathbb{H}$ (but not $\mathbb{O}$). 

*Field extensions (like $\mathbb{R}[x]$ or $\mathbb{R}(x)$ as $\mathbb{R}$-vector spaces, or $\mathbb{Q}[\sqrt{2}]$ as a $\mathbb{Q}$-vector space)

*The algebra of $n\times n$ matrices $\mathbb{R}^{n\times n}$

*The algebra of continuous functions $C(X)$ for any topological space $X$ (you can just consider $\mathbb{R}$ or $[0,1]$ if you're not comfortable with topological spaces)

*Lie Algebras
All of these things are useful and some have entire fields of study devoted to them. 
