$LaunchDirectory gives the directory in which the executable for the Mathematica kernel program being run resides.
$LaunchDirectory returns the full name of the directory as a string.
$LaunchDirectory gives the directory for the actual Mathematica kernel executable, not the directory for scripts such as math that call it.
After installation on a hard disk, $LaunchDirectory is typically the Kernel/Binaries subdirectory of $TopDirectory.
See The Mathematica Book: Section 2.11.4.
See also: $TopDirectory, $CommandLine, $InitialDirectory, $InstallationDate.