However, you can setup postgresql or SQL Express and create a connection to them in ArcCatalog. Then you can import/create feature classes and connect to and view in desktop -- but no editing with desktop. I'm pretty sure you have to have a license to enable geodatabase setup in any db (which allows you full functionality). Esri is moving towards direct connect to databases -- defaults to direct connect now in 10.1. You would need to install postgresql on the client if using it.
Traditionally, there's roughly 2 parts to SDE -- all the geodatabase tables that enable versioning, topology, etc. and then a seperate service on the database server for connections (ie port 5151 stuff, you didn't use the normal database connection). Esri now says to just use direct connections unless there is a reason to create the SDE service, e.g. some clueless network guy says no direct connections to an Oracle db.
All that said... there are services everywhere to add to the client -- which again is kind of the newer paradigm for working with spatial data and maps -- not that an app connecting to a database is outdated. Esri has hundreds of online services specifically for development. ArcGIS Online has thousands from different people. Open source WMS, WFS, KML everywhere.