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I've seen a few references to higher D (4D+) variants on here, but nothing
beyond stacking boards, which is of course just stretching 3D space. 

Naturally, its not particularly practical to implement such games at
present-not too many 4D+ GUIs around as yet. However there is another
preceding and more important core approach to play (or at least simulate) a
higher D chess game, or more intricate chess variants in general. More on
that later...

Its pretty interesting (for me at least) to think about how such a
fundamental and well balanced abstract concept as chess transfers into
higher dimensions. A 6D Chess game for example would, as far as I can make
out, have 4,096 pawns/pieces for either side, 6 straight simple sliders, (6
choose (compound level)) compounds for each level, (eg. (6 choose 2)=30
double compound pieces, still just the one, extremely powerful, queen), and
(((range*number of orthonogal axes acted simultaneously upon)-2) leapers
(eg 2 2-range leapers acting across 3 axes, the 1:2:2:0:0:0 and
1:1:2:0:0:0).

Quite a bit more going on than FIDE Chess...

But how on earth would a mere human get any kind of appreciation for such a
game? A game with an incomprehensibly large branching factor, that would
take a solid year to play?

This is where I segway into the concept of macrocommands. :)