To clarify:
You don't need ArcSDE enterprise to create a geodatabase from the database server node in ArcGIS Desktop. In fact, you can't create an enterprise ArcSDE geodatabase from ArcGIS Desktop at all.
To create a geodatabase on a SQL Server Express instance through the database server node in Desktop, you need:
-A supported version of SQL Server Express
[INDENT]You can run the SQL Server Express installation wizard provided with ArcGIS Desktop or ArcGIS Server Workgroup to install this.[/INDENT]
-To run the portion of the installation wizard to enable the SQL Server Express instance to store geodatabases, which writes a key to the Windows registry
[INDENT]When you create a geodatabase on a database server, ArcGIS looks for this registry key. If if can't find it, you won't be able to create a geodatabase.[/INDENT]
-A connection to the database server from an ArcEditor or ArcInfo level ArcGIS Desktop client
[INDENT]If your ArcGIS Desktop license isn't set to ArcEditor or ArcInfo, the New Geodatabase command will be inactive.[/INDENT]
-To be logged in to the client computer as an administrator on that particular database server
[INDENT]A database server administrator is added at the time SQL Server Express is installed. That initial administrator can add other user logins to the database server administrator role if needed. Only database server administrators have the privileges in the SQL Server Express instance required to create a geodatabase.[/INDENT]
