Hausdorff Dimension Infinity

What are some examples of non-trivial metric spaces that have Hausdorff Dimension of infinity?

I could only think of $$\mathbb{R}$$ with the discreet metric.

Take the separable Hilbert space of infinite dimension $$\ell^2=\{(a_n)_{n\in\mathbb{N}}\subseteq \mathbb{R}:\sum_{n\in\mathbb{N}}a_n^2<\infty\}$$
• +1 Did you know that the subset of $\ell^2$ with all rational coordinates has Hausdorff dimension 1? One of my favorite results of Erdos. – Mark McClure Nov 27 '18 at 0:23