Can a single chart be used to summarize three categorical variables, much like a Marimekko Chart can be used to summarize two? Obviously, a set of Marimekko Charts could be displayed in a small multiple fashion, but I am looking for a single chart instead. And if 3D rendering is necessary, it should not require any rotation or zooming for the full dataset to be comprehended at a glance. And to be clear, the three categorical variables are the independent variables, while the chart is depicting them with a single dependent variable, which is assumed to be numerical.
Marimekko plots are a subset of mosaic plots. You can interweave in third or fourth variable. There is a nice example on wikipedia for the titanic data. The vcd package in R can make these for you, and a variation called a double-decker plot. Also the package prodplots adds a formal structure for defining plots for categorical variables. Another package called extract has a variety of tools for categorical data. There are also other approaches like hammock plots, an adaptation of parallel coordinate plots, are available in ggparallel.