Algebraically, What Does $\Bbb R$ get us? In terms of the basic algebraic operations -- addition, negation, multiplication, division, and exponentiation -- is there any gain moving from $\Bbb Q$ to $\Bbb R$?
Say we start with $\Bbb N$:  $\Bbb N$ is closed under addition and multiplication.  But then we decide we'd like a number system that's closed under negation as well, so we construct $\Bbb Z$.  Great.  But then we decide we'd like to extend this number system further to be closed under division and so we construct $\Bbb Q$.  The next step is closure under exponentiation - but when we construct that number system, we don't get $\Bbb R$, we get a subset of $\Bbb C$ which I'll call $\Bbb Q_{\exp}$.
Now clearly when constructing $\Bbb R$ from $\Bbb Q$ we do gain completeness, but our gain is then analytic, not necessarily algebraic.  Do we gain any algebraic advantage in constructing $\Bbb R$ from $\Bbb Q$ similar to what we get at each of the other constructions I mentioned?

As Asaf mentioned, you cannot obtain $\mathbb{R}$ as a completion of $\mathbb{Q}$ under these operations. However, there are still algebraic things to be said about $\mathbb{R}$. 
For example, every irreducible polynomial over $\mathbb{R}$ has degree less than or equal to two, which is trivial to check since $\mathbb{C}$ is algebraically closed, contains $\mathbb{R}$ and $[\mathbb{C}:\mathbb{R}]=2$. 
This implies, for instance, that $x^4+1$ can be factored over $\mathbb{R}$. And the factorization is $(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)$.
