Suppose $X_{0} = X$ is a complex space of dimension 2 with divisor $p_{0} \in X_{0}$. We can construct the blow-up, $X_{1}$ of $X$, which comes with a blow-down map $X_{1} \to X_{0}$. Suppose that there is a distinguished point $p_{1} \in X_{1}$. Then we can blow up again : $X_{2} \to X_{1}$. If this process continues indefinitely, we get a directed system of spaces:
$...\to X_{n} \to X_{n-1} \to ... \to X_{0}$.
In this case, it makes sense to consider the inverse limit $\lim_{\leftarrow} X_{n}$.
My question: Does such a space exist, and if so is there a nice description of it (or if possible a construction)?
Blowing up a point on a curve doesn't do anything, so I will assume you are talking about surfaces or higher dimensional varieties.
In that case, this inverse limit exists, but only as an inverse limit (although see edit). It's not a projective variety, for example because its cohomology group $H^2(X,\mathbf Z)$ is infinite dimensional. But people still work with this object! Some names for this (or related objects) are Zariski-Riemann space and bubble space.
I don't know what it means to "compute" this space.
Edit: According to Wikipedia, the Zariski--Riemann space is a locally ringed space. So that's a geometric category, containing the category of schemes, in which this inverse limit can be realised.