Probably my favorite are the Myriahedral projections, basically the Dymaxion map, but with ridiculous number of faces. In his paper he also unfolds a fractal.
I like Cahill-Keyes: http://www.cartotalk.com/index.php?showtopic=9406 (d3.js animation included).
I think Healpix is included in the tool at the bottom. Also check out the last few in the list which are similar to Dymaxion.
There's a few map projections I really like that don't have most of these problems: HELAPIx and Dymaxion map.
HELAPIx is designed to have pixels of the same area, and Dymaxion is designed to have the least distortion.
One of the main reasons it's so popular is that is preserves shape, angle and north is always up. This means you can create a tiling map server and use the same tiles at all scales.
It is possible, but not trivial:
https://www.vis4.net/blog/posts/no-more-mercator-tiles/ (comments are also informative, particularly Tom MacWright's)
Are there any online maps that aren't Web Mercator? Google Maps, Here.com, OpenStreetMap, Bing Maps, MapQuest are all Mercator-only, with no easy option to switch.
I don't yet have a very good understanding of the whole EPSG code system yet either. :s
Is there some nice explanation of how the projections between various EPSG codes work? I converted one projection into another but in QGIS or in python pyproj, but except for the very basics I have no idea what is going on.
nice, one of the best, interactive visualizations of the problems of mercator maps i've seen
>To remedy the horizontal stretching, the Mercator Projection stretches the globe out in the vertical direction an equal amount to the stretching in the vertical direction.