To answer the LPI question I think that there isn't any actual directory and that this is a trick question! Linux has no mandatory directories that must be present, it can be installed into any directory structure(s) one chooses, it's configurable.
I did dig up this page, titled: 4. Linux Directory Hierarchy: Oriented to the Software Parts. It has this to say:
So let's summarize what the FHS has to say about Linux directories:
Linux system directories
Directory for the executables that are accessed by all users (everybody have this directory in their $PATH). The main files of your Software will probably be here. You should never create a subdirectory under this folder.
Like /usr/bin, but here you'll find only boot process vital executables, that are simple and small. Your Software (being high-level) probably doesn't have nothing to install here.
Like /usr/bin, but contains only the executables that must be accessed by the administrator (root user). Regular users should never have this directory in their $PATH. If your Software is a daemon, This is the directory for some of executables.
Like /usr/sbin, but only for the boot process vital executables, and that will be accessed by sysadmin for some system maintaining. Commands like fsck (filesystem check), init (father of all processes), ifconfig (network configuration), mount, etc can be found here. It is the system's most vital directory.
Contains dynamic libraries and support static files for the executables at /usr/bin and /usr/sbin. You can create a subdirectory like /usr/lib/myproduct to contain your helper files, or dynamic libraries that will be accessed only by your Software, without user intervention. A subdirectory here can be used as a container for plugins and extensions.
Like /usr/lib but contains dynamic libraries and support static files needed in the boot process. You'll never find an executable at /bin or /sbin that needs a library that is outside this directory. Kernel modules (device drivers) are under /lib.
Contains configuration files. If your Software uses several files, put them under a subfolder like /etc/myproduct/
The name comes from "variable", because everything that is under this directory changes frequently, and the package system (RPM) doesn't keep control of. Usually /var is mounted over a separate high-performance partition. In /var/log logfiles grow up. For web content we use /var/www, and so on.
Contains the user's (real human beings) home directories. Your Software package should never install files here (in installation time). If your business logic requires a special UNIX user (not a human being) to be created, you should assign him a home directory under /var or other place outside /home. Please, never forget that.
You may think is a bad idea to break your Software (as a whole) in many pieces, instead of keeping it all under a self-contained directory. But a package system (RPM) has a database that manages it all for you in a very professional way, taking care of configuration files, directories etc. And if you spread your Software using the FHS, beyond the user friendliness, you'll bring an intuitive way to the sysadmin configure it, and work better with performance and security.
So at least according to this later section to the FHS these are some directories that one could consider as loosely mandatory, but Linux doesn't have to follow the FHS. Even the LSB (Linux Standards Base) is only a guide, not a ridgid specification.
The only thing that Linux tries to adhere to is POSIX compliance, and POSIX makes no conveyances such as this when it comes to a specific variant of UNIX.
 The following material is excerpted from
Graham Glass and King Ables, Linux for Programmers and Users,
Pearson Prentice-Hall, 2006. ISBN 0-13-185748-7. p 4-15