Hausdorff Dimension Infinity

Question

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

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

Answer 1

Take the separable Hilbert space of infinite dimension $$ \ell^2=\{(a_n)_{n\in\mathbb{N}}\subseteq \mathbb{R}:\sum_{n\in\mathbb{N}}a_n^2<\infty\} $$