MangoMap: 

GISCloud: 

MapBox: 

GeoCommons

Open Geo Suite

And of course Google Earth

https://mangomap.com/arcgis-online-alternative

http://www.giscloud.com/

http://mapbox.com/

http://geocommons.com/

http://opengeo.org/products/suite/cloud/

Google maps api or the mapping abilities included in google drive/docs may be sufficient.

After trying Mango Maps, open Geo Suite, and Google Earth, I went back to ArcGIS online. While I really liked open Geo Suite (so much you can do!), I don't have access to a server for public use of the maps.  Thanks for your input, Stu and Samuel.