Ok, this might be a very stupid question but I couldn't find anything online so I just ask.
If we blow-up a point in $\mathbb A^n$, we obtain something inside $\mathbb A^n\times \mathbb P^{1}$, since a point is defined by $k=n-1$ equations and $n-(n-1)=1$. More generally, if we blow-up anything defined by $k$ equations, the blow-up lives inside $\mathbb A^n\times \mathbb P^{n-k}$.
There may be a lot of issues with viewing the empty set as a subvariety but still, can we define (as a pathology) the blow-up of $\mathbb A^n$ along the empty set?
The blowup of a scheme $X$ with center in a complete intersection of codimension $k$ can be embedded into $X \times \mathbb{P}^{k-1}$ (in particular, the blowup of a point in $X$ of dimension $n$ embeds into $X \times \mathbb{P}^{n-1}$, not into $X \times \mathbb{P}^{1}$).
On the other hand, the ideal of the empty set in $X$ is $$ I = \mathcal{O}_X, $$ hence its blowup is $$ \operatorname{Proj}_X( \oplus_{k=0}^\infty I^k) = \operatorname{Proj}_X(\mathcal{O}_X[t]) = X \times \mathbb{P}^0 = X. $$