I'm looking for an alternative to Google-maps with all the richness of their API but more open. Does such a thing exist?
About...
OpenLayers makes it easy to put a dynamic map in any web page. It can display map tiles and markers loaded from any source. MetaCarta developed the initial version of OpenLayers and gave it to the public to further the use of geographic information of all kinds. OpenLayers is completely free, Open Source JavaScript, released under a BSD-style License.
OpenStreetMap is, of course, the obvious answer. There are various APIs for managing the data in the database, and there are various APIs for getting maps onto your webpages, such as OpenLayers, Mapstraction, or Staticmaps.
If you are in the UK you can use the Ordance Surveys' OS Open Data maps.
Consider this article: http://www.alistapart.com/articles/takecontrolofyourmaps
Here are a few options suggested by the article:
1-OpenLayers
2-TileCache
3-Mapnik
4-QGIS
The article explains (theoretically) how to create your own software stack that will mimic google maps, but remain under your complete control.
You might not need all the layers of software for your project, but you may find the article helpful regardless.