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?



Triply graded spectral sequence?
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.


