I never really was one to severely restrict my palette, but some basic color mixing of three paints wound up mimicking the hues of a many of my standard colors. The 3 paints were Primary Yellow, Primary Red Magenta, and Primary Cyan (Maimeri Brera Acrylics).
Below are closer looks at the paints and the name of the tubed color to which it could be akin:
Primary Yellow is straight from the tube. The yellow-green would be akin to Phthalo Yellow Green, only more transparent.
The secondary green: Phthalo Green Yellow Shade
Blue Green: Phthalo Green Blue Shade
Primary Cyan straight from the tube. By the time I finished the color wheel, I realized I skipped Turquoise and Middle Blue. The turquoise looks like Phthalo Turquoise, which is just Phthalo Blue + Phthalo Green BS. The middle blue could mimic a deep cobalt blue to a more Prussian Blue hue. If I would have added less Primary Magenta, I could've come out with a more Ultramarine blue hue.
Blue Violet: A deeper Dioxazine purple?
Violet: Ultramarine Violet. Maybe even a deep Cobalt Violet hue.
Red Violet: Quinacridone Magenta or Thio Violet.
Primary Magenta straight from the tube. With a 1/3 as much Primary Yellow, a middle red similar to Naphthol Red Light was accomplished.
Red Orange (bottom left) - Perinone Orange.
Secondary Orange: Benzimida Orange
Yellow Orange: Indian Yellow / Gamboge.
One limitation of this palette for me would be lack of opacity in any of the paints, especially on the left side of the paradigm. While one can get Cad Red / Orange / Yellow hues, none of them will be opaque enough. Still, for those that paint in mostly transparent colors, there are tremendous possibilities with this 3-paint palette.
For those that use it, you could even throw Ivory Black into the mix as a greying agent and achive more!