One thing to keep in mind here is the MS Access product is much more than just the raw database engine. It provides a full application development platform, including form and menu designer, client application language and environment (VBA), and report designer. When you take all those things together, MS Access really has no peer.
But for the scope of this question, we're concerned with the raw database engine. With that in mind:
SQLlite,
Firebird,
VistaDB (not free),
SQL Server Compact Edition (not Express)
and now SQL Server LocalDB
all come to mind.
Another thought: while the original question does ask about desktop databases, its likely some people will land here looking for a database to use with a web site. It's important to remember that these are all in-process databases, and as such are rarely if ever appropriate for use on the web. If you want to build a web site, where it's common to need to support significant concurrent access, you generally want a database server engine, like MS SQL, Postgresql, MySQL, Oracle, or their brethren. At the same time, those server engines are rarely if ever appropriate for a single-user desktop application.
