The ArcGIS Sever Development Blog says that the ArcGIS JavaScript API is now available to the public. As was mentioned before, it is freely available for anyone to use. You can of course also use the extenders for Virtual Earth and Google Maps if the ESRI JavasScript client isn’t what you are looking for.

The ESRI ArcGIS Server Resource Center has all the information about getting started with the API. You don’t need your own ArcGIS Server install to get started, you can just use ESRI’s sample servers.