To get the best possible answer to this question I combine all given answers on this question.
Alternatives to google maps api:
MapTiler Cloud:
https://cloud.maptiler.com/
Bing Maps (Microsoft Maps) API:
https://docs.microsoft.com/en-us/bingmaps/
MapQuest:
http://developer.mapquest.com/
(provided by TuxGeek)
Mapjam:
http://mapjam.com
OpenStreetMap API:
http://wiki.openstreetmap.org/wiki/API_v0.6
Osmdroid:
http://wiki.openstreetmap.org/wiki/Osmdroid
OpenStreetMap Android
http://wiki.openstreetmap.org/wiki/Android
(provided by AGrunewald)
GraphHopper Directions and Matrix API
https://graphhopper.com/#directions-api
(provided by karussell)