Multi-Datacenter design for SQL Server
At this point, I guess SQL Server with an AlwaysOn cluster will
  satisfy HA, but what about read/write?


I think this SQL Server 2012 whitepaper will be a good read for you at this point to get a feel for what's possible with the two different features (Availability Groups and Fail-over Clustered Instances) that come under the AlwaysOn brand name.


  Deploy a datacenter in every region you want and every database will
  contain the same information with read/write and HA capabilities


You ask about the read/write capabilities and I think this is where you will be disappointed. Even with a combination of local FCIs with a cross site AG it is only ever possible to have one node as the PRIMARY (writable) node. The other nodes can act as read-only nodes to allow you to off load reads and backups.

From my experience with building systems like this you really want to keep things as simple as possible. For this reason I wouldn't recommend trying to extend an AlwaysOn FCI + AG to more than 2 sites. I would also stay clear of bidirectional transnational replication as this has a tendency to break.

Without more information it is difficult to recommend a solution for you. Some very general topics to consider would be; a caching layer in your app at each site to help with the latency or a SQL Server Azure deployment.

After the update to the question:

I really wouldn't recommend you to use SQL Server in that way. Bidirectional replication between 2 nodes is bad enough. >2 nodes plus the cost of all those Enterprise Editions of SQL Server just sounds like a bad idea. Maybe take a look at what Azure has to offer.