As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this topic?

Well, there is an eponymous spectral sequence in my thesis (still never published in full, but there is an announcement, stuff about it in Ravenel's book, and papers by Tangora and others). Quite generally, take a connected graded algebra $A$ over a field $k$, filter it for example by the powers of its augmentation ideal $IA$ (the elements of positive degree). Then the associated graded algebra $E^0A$ is bigraded. The filtration leads to a spectral sequence that converges from $Ext_{E^0A}(k,k)$ to $Ext_A(k,k)$. There is a homological grading and a bigrading from $E^0A$ that give a trigrading.

Incidentally, in the opposite direction, Bockstein spectral sequences of spaces are monograded.

Of course this has been studied. You need but google "trigraded complex", and much wisdom will be found. OK, some wisdom, notably Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres, and a very cute presentation by Noah Forman on

Graham Denham. The combinatorial laplacian of the tutte complex. J. Algebra, 242(1):160-175, 2001.

and related matters.

As others have said, there are plenty of examples of spectral sequences that have a third grading such that each $d_r$ preserves the grading up to a shift depending only on $r$. This is all fairly straightforward.

However, there are some more interesting questions along the same lines. When Ravenel was trying to disprove the Telescope Conjecture he had certain spectra with a doubly-indexed filtration, say $X_{ij}$. Given a slope $m>0$ one can define a singly-indexed filtration by $F_kX=\bigcup_{i+mj\geq k}X_{ij}$, and using this we obtain a spectral sequence converging to $\pi_*(X)$ (the "localised parametrised Adams spectral sequence"). Ravenel's approach was to analyse how this changes when we vary $m$. For a long time he was asking whether there was some kind of spectral-sequency gadget that incorporated all values of $m$ at the same time. I don't think anyone ever had a satisfactory answer to that.

J. L. Verdier dissertation (written in the 60s), pre-Ravenel's book, covers multiple-graded complexes. It was reprinted "recently"

Des catégories dérivées des catégories abéliennes. (French. French summary) [On derived categories of abelian categories] With a preface by Luc Illusie. Edited and with a note by Georges Maltsiniotis. Astérisque No. 239 (1996), xii+253 pp. (1997).

Calcul des variations et topology algébrique, page 82), which may be seen as a "trigraded spectral sequence". I'm aware that by googling "trigraded spectral sequence" one gets a lot of results, but it's hard to find a place where main definitions and basic results of the theory (cont.) $\endgroup$ – Giovanni Moreno Apr 10 '12 at 11:46