Let $(X,d)$ be a metric space and $A\subseteq X$ be compact.

Since every finite set is doubling then the collection $\mathcal{A}$ of doubling subsets of $A$ cannot be empty. My question is, how large can the following be $$ \sup_{\tilde{A}\in \mathcal{A}}\,d(A,\tilde{A}) ? $$

Alternatively, is there a way to quantify how far a metric space is from being doubling?