Let $\mathcal{E}$ be a $n$-dimensional Euclidean space with full dimensional Lebesgue measure $\mathcal{L}$. For $(X,x) \in 2^{\mathcal{E}} \times \mathcal{E}$, upper and lower densities of $x$ in $X$ are: \begin{eqnarray} d^{+}(X,x) := \limsup_{r \downarrow 0} \frac{\mathcal{L}(X \cap B(x,r)))}{\mathcal{L}(B(x,r))} \\ d^{-}(X,x) := \liminf_{r \downarrow 0} \frac{\mathcal{L}(X \cap B(x,r)))}{\mathcal{L}(B(x,r))} \end{eqnarray} where $B(x,r)$ are closed balls.
Since $X \cap B(x,r) \subseteq B(x,r)$ then $\mathcal{L}(X \cap B(x,r)) \leq \mathcal{L}(B(x,r))$ it is reasonable to say that $d(X,x)\leq 1$ is always true. However, Mattila "Geometry of sets and measures in Euclidean spaces" at p. 90 does not exclude the existence of pairs $(X,x)$ for which density can be bigger than $1$. Specifically, in the proof of Thm. 6.2 he proves that the set of points with density bigger than $1$ has measure zero without proving that it is empty.
Whereas for "nice" sets upper and lower densities exist, coincide and cannot be bigger than $1$, an arbitrary set may have points for which upper and lower densities do not agree, are bigger than $1$ or are not defined.
Of course, the existence of such sets and points should be of a rather "pathological" nature but I would be interested to know if and to what extent it is possible to characterize the sets $X$ for which there are special points in the above sense.