I am a basic level user. Can I use any DBMS types with ESRI? I did find out that to use postgres I had to have at least a standard. I am assuming across the board ESRI has entry to this functionality at the standard... more expensive level.
Depending on the usage you will probably need a Standard license (at least for editing an enterprise gdb). More details on what you can do with each license can be found here:
Retrieving data ...