  In this article we construct what we call a higher spectral sequence for any
chain complex (or topological space) that is filtered in $n$ compatible ways.
For this we extend the previous spectral system construction of the author, and
we show that it admits considerably more differentials than what was previously
known. As a result, this endows the successive Leray--Serre, Grothendieck,
chromatic--Adams--Novikov, and Eilenberg--Moore spectral sequences of the
author with the structure of a higher spectral sequence. Another application is
a universal coefficient theorem analog for spectral sequences.
