Inspired by the wonderful World Map generator and the few different projections already implemented on the Paleoenvironment map I dove deeper into the wonderful world of map projections for OpenLayers.
The most straightforward option for me was to use PROJ4JS. Unfortunately it is not straightforward for all projections. It requires a code from the EPSG database, together with the PROJ description. Some projections, although included in PROJ4JS, are not in EPSG and therefore I could not plot those. For instance the awesome "Quadrilateralized Spherical Cube". Then there is the problem which parameters to set. Some parameters in the description result in incomplete maps and usually the extent and world extent are not included in PROJ4. Additional issue is that the extents depend on the unit used for the coordinates. One thing I would have liked to add, but did not get to work was an option to interactively set the central meridian and other parameters
Anyway, it was fun playing around. Have a try and expect some unexpected results when switching between projections. If that happens, you can always restart by reloading the page.
As an additional fun item, I also made a Fuller-like projection work: that's here