I recently learned about spectral sequences (which I am still trying to understand). In class we defined a (co)homologically graded exact couple (of $A_p^n$s and $C_p^n$'s with vertical arrows in the $A$ columns) to be a certain grid of exact sequences, which given some convergence assumptions, allows us to calculate the direct limit over $p$ of the $A_p^n$ in terms of $E_\infty (C_p^n)$ (this is a pretty rough sketch). I know inverse limits in general behave worse than direct limits, but in some nice cases, is there something like spectral sequences to compute inverse limits instead? There are lots of interesting things in number theory that are inverse limits is why I am wondering about this.
I think that the best thing to do when you want to start learning about spectral sequences is to do computations, see this answer for an outline: Reference for spectral sequences
(I really believe what I said there, it is not a plug.)
I was going to recommend a paper, but I looked at your profile and that drasticly changes my advice. Ask Mike Boardman! Or at least try to trace through the first part of his conditionally convergent spectral sequences! That paper is really flipping cool. It is not easy, but it isn't really hard. Also, you can probably skip the alternative descriptions (the weird big formulas in the first section) of the $E_r$ terms for now.
Seriously, you want to read that paper.
Edit: I feel bad for not explicitly answering your question, but all I would do is look up something in that paper and mess it up trying to copy it down.