$\DeclareMathOperator{\Ext}{Ext} \newcommand{\A}{\mathcal{A}} \newcommand{\F}{\mathbb{F}} \newcommand{\Sq}{\mathrm{Sq}} \newcommand{\Z}{\mathbb{Z}}$Here's another another example that's worth considering. The method I'll suggest for computing the differentials is a lot hairier and more technically complex than the others, but it is a generic method, and it's very rewarding to watch work. I should immediately mention this that I first heard of this from Mark Behrens, who in turn said he heard of it from Mikes Hill and Hopkins. I don't know whether the chain stops there or if it continues on.

Tilman Bauer mentioned a nice example of a spectral sequence, the homotopy fixed point spectral sequence arising from the complex conjugation $C_2$-action on complex $K$-theory $KU$: $$E_2^{*, *} = H^*(C_2; \pi_* KU) \Rightarrow \pi_* KU^{hC_2} \cong \pi_* KO.$$ The $E_2$ page of this is simple enough to compute; it ends up looking like $E_2 = \Z[\eta, [\beta^2]^{\pm}] / (2 \eta)$, where $[\beta^2]$ is a class in degree $(4, 0)$ coming from $\beta^2 \in \pi_* KU = \Z[\beta^\pm]$, and $\eta$ is a class in degree $(1, 1)$ coming from the nontriviality of complex conjugation on the Bott bundle $\beta$. For good measure, here's a picture of this spectral sequence:

What's much less obvious to compute is the first nontrivial differential I've drawn in: $$d_3 [\beta^2] = \eta^3.$$ The way people usually see this, as far as I've heard, is to independently identify $KU^{hC_2}$ with $KO$ and then notice that this differential must exist for real Bott periodicity to be true.

But that isn't what you were asking for, as you were hoping to compute differentials more manually. Here is a different route to producing this differential. As $KU$ is a ring spectrum, it comes with a unit map $\mathbb{S}^0 \to KU$, which is equivariant for the trivial $C_2$-action on $\mathbb{S}^0$. The naturality of fixed point constructions then begets a map of spectral sequences $$\begin{array}{ccc} H^*(C_2; \pi_* \mathbb{S}^0) & \Rightarrow & \pi_* (\mathbb{S}^0)^{hC_2} \\ \downarrow & & \downarrow \\ H^*(C_2; \pi_* KU) & \Rightarrow & \pi_* KU^{hC_2}. \end{array}$$ This becomes useful after making an identification: $(\mathbb{S}^0)^{hC_2}$ for the trivial action is given by the function spectrum $F(BC_2{}_+, \mathbb{S}^0)$, i.e., for the Spanier-Whitehead dual spectrum $D \Sigma^\infty_+ \mathbb{R}\mathrm{P}^\infty$.

Now recall the cell structure of the top bits of $D \Sigma^\infty_+ \mathbb{R}\mathrm{P}^\infty$. Diagrammatically, this is drawn as $$\cdots \overbrace{\bullet - \bullet \phantom{{}-{}} \bullet} - \bullet \phantom{{}-{}} \bullet,$$ where each $\bullet$ denotes a cell (with the rightmost one in dimension 0), each $-$ denotes the multiplication-by-2 map, and the brace denotes the element $\eta$ in the stable stem $\pi_1 \mathbb{S}^0$. This cell structure is actually what dictates the map $H^*(C_2; \pi_0 \mathbb{S}^0) \to H^*(C_2; \pi_0 KU)$ in the spectral sequence: on the $E_1$-page, each cell is represented by a $\Z$ and sent isomorphically to a corresponding $\Z$ in the $E_1$-page for $\pi_* KU^{hC_2}$. The multiplication by $2$ attaching maps become $d_1$-differentials; the action of $d_1$ on the cell in dimension $-s$, corresponding to $H^s(C_2; \pi_0 \mathbb{S}^0)$, is given by multiplication by the degree of the attaching map, and what survives is sent to the elements $1$, $\beta^{-2} \eta^2$, $\beta^{-4} \eta^4$, ... in the $E_2$-page for $\pi_* KU^{hC_2}$.

The exciting (and final) observation is that this same procedure determines the $d_3$-differential: it acts on $H^2(C_2; \pi_0 \mathbb{S}^0)$ (i.e., the $(-2)$-cell representative) by mapping to $\eta$ times $H^4(C_2; \pi_0 \mathbb{S}^0)$ (i.e., the $(-4)$-cell representative). Pushing this forward into the spectral sequence for $\pi_* KU^{hC_2}$ begets the differential $$d_3(\beta^{-2} \eta^2) = \beta^{-4} \eta^4 \cdot \eta = \beta^{-4} \eta^5.$$ Translating this differential around using the Leibniz structure recovers the differential I claimed at the start.

These are a lot of fancy words, and proving all the relationships I've claimed in this response is not an easy task, but the end result is very neat! It's also a very general technique: studying these equivariant cells in other $G$-spectra (e.g., some variants of tmf) allows you to produce scores of interesting differentials that you didn't know about before. This specific example is not something I would push on someone "opening the black box" for the first time, but maybe the second or third hundredth time it seems like a fine idea.

In the meantime, the message to take away is that importing differentials by naturality from a filtration spectral sequence which you understand well is a powerful tool when the filtration complex for your favorite spectral sequence is not so easy to write down or to compute with directly.

Think of the specific example as something to look forward to after digesting the rest of the responses to this question.

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