I'll follow the definition given here:
The doubling constant of metric space M is the maximum, over all balls b in the metric space M, of the minimum number of balls needed to cover b, using balls with half the radius of b. The logarithm of the doubling constant is the doubling dimension of the space.
So far I've seen that Euclidean $(\ell_2)$ space $\mathbb R^d$ has doubling dimension $\Theta(d),$ and also several results which assume a metric space of finite doubling dimension, but as of yet I haven't been able to find a counterexample to this, or in other words a metric space with unbounded doubling dimension.
What are some examples of this?
Consider $\Bbb Z$ with the metric $$ d(x,y) = \begin{cases} 0 & \text{if } x=y,\\ 1 &\text{if } x\neq y. \end{cases} $$ Then the open ball $B(0,2)$ is $\Bbb Z$, and each ball of radius $1$ $B(x,1)$ is the singleton $\{x\}$. It then takes infinitely many of such balls to cover $B(0,2)$, and the doubling dimension of $(\Bbb Z,d)$ is infinite.