One example is a construction that is often used in the passage from smooth projective varieties to arbitrary varieties. There are various variants of this:

 - For a proper variety $X$, take a hypercovering $X_\bullet \to X$ by smooth projective varieties (using alterations plus the inductive procedure of SGA 4$_{\text{II}}$, Exp. V$^{\text{bis}}$, §5). Then there is a hyperdescent spectral sequence
$$E_1^{p,q} = H^q_{\text{ét}}(X_p,\mathbf Q_\ell) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
 - For a smooth variety $X$ with a smooth compactification $\bar X$ whose complement $Z = \bar X \setminus X$ is a simple normal crossings divisor $\bigcup_{i \in I} Z_i$, there is an excision spectral sequence
$$E_1^{p,q} = \bigoplus_{\lvert J \rvert = p} H^q_{\text{ét}}\Big(\bigcap_{i \in J} Z_i,\mathbf Q_\ell\Big) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
 - These methods can be combined: as I explain in [vDdB20, Thm. 6.6], for an arbitrary separated finite type $k$-scheme $X$, one may construct a simplicial variety $X_\bullet$ with smooth projective components $X_i$ and a spectral sequence
$$E_1^{p,q} = H^q_{\text{ét}}(X_p,\mathbf Q_\ell) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
It has been suggested to me that one should be able to prove this using Voevodsky motives, but I don't know a reference. (My paper is written in the more classical language of Chow motives.)

There are also versions for singular cohomology if $k = \mathbf C$. For instance, the first example above is used to *define* the mixed Hodge structure on the cohomology of an arbitrary $\mathbf C$-variety.

Each of the above examples degenerates on the $E_2$ page for weight reasons: any $E_r^{p,q}$ is pure of weight $q$, so the differentials are forced to be $0$ once the source and target live in different rows. The observation that you only need the $E_1$ differentials is one of the key points of [vDdB20].

(To make this weight argument precise, either reduce to a finite type situation over $\operatorname{Spec} \mathbf Z$ and use Frobenius eigenvalues, or (if $\operatorname{char} k = 0$) reduce to a finite type situation over $\mathbf Q$, choose an embedding into $\mathbf C$, and use Hodge theory.)


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**References.**

[vDdB20] <cite authors="van Dobben de Bruyn, Remy"> R. van Dobben de Bruyn, [*The equivalence of several conjectures on independence of $\ell$*](https://doi.org/10.46298/epiga.2020.volume4.5570). Épijournal Géom. Algébrique **4**, Art. nr. 16 (2020). [ZBL1460.14053](https://zbmath.org/?q=an:1460.14053).</cite>