What vector formats can GDAL write that ArcGIS for Desktop can read without the need for its Data Interoperability extension?
I do not plan to use either Personal Geodatabases or Shapefiles
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ArcGIS 10.2 supports SQLite, and appears to be imminent/available, so SQLite could be a good option for you.
You can connect from ArcGIS to an SQLite database to create maps and perform spatial analysis on your data.
You connect directly to the SQLite database file from your ArcGIS client.
One good option is storing your data in a spatial database like PostGIS.
You can read and write to this database from most OpenSource Applications. ArcGIS 10.0 and above can read this data directly.
If you need to write to PostGIS from ArcGIS, that is also possible by using third party extensions.
A promising vector format for this purpose is SQlite/Spatialite. ArcGIS users can access Spatialite files through AmigoCloud's GDAL/OGR Plugin for ArcGIS.
If your vector data is point format than you could use .csv files to go between Arc and open source applications (as long as you have a x/y or lng/lat columns).