In this question I previously asked how to think about the motivic complex $ \mathbf{Z}(1)_{\mathcal{M}}$ , whose Zariski hypercohomology should morally be the “singular cohomology” $ H^*((-)\wedge S^{2n},\mathbf{Z})$ , or the “singular cohomology of a pair” $ H^*((-)\times\mathbf{P}^n_k,(-)\times\mathbf{P}^{n-1}_k)$ .

Denis Nardin’s answer explains a suggestive analogy, and here is a followup.

It happens that several motivic cohomology groups are infinitely generated, even if $ X$ is a smooth projective variety over a field (even for $ k = \mathbf{Q}$ and $ X = \text{Spec}(k)$ ).

How to think about this? Is there an analogy with infinite generatedness of the singular cohomology of a pair, for instance?

In other words, what should be the “moral reason” why this infinite generatedness occurs? Any analogy coming from algebraic topology would be greatly clarifying.