A rather curious idea popped into my head lately, as will be detailed in the following blog post.
The original Graham's number was defined as so:
"The smallest number of dimensions a hypercube can have, so that it is impossible to colour (or "label") all the edges of the hypercube red or blue, without colouring all the six edges between any four coplanar vertices the same colour."
Now, we call this a "Second order" Graham's number, because we used two colours, red and blue.
In general, an nth-order Graham's number (noGn, pronounced "Enogen") is defined identically, only that we replace "red or blue" with n colours.
I might be ever so slightly wrong in the definition of Graham's number, so if I am, please put it to my notice.