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You want to back up as much as possible. The major
exception is software that can be easily reinstalled,
but even they may have configuration files that it is
important to back up, lest you need to do all the work to
configure them all over again. Another major exception is
/proc filesystem; since that only
contains data that the kernel always generates automatically,
it is never a good idea to back it up. Especially the
/proc/kcore file is unnecessary, since it
is just an image of your current physical memory; it's pretty
large as well.
Gray areas include the news spool, log files, and many
other things in
/var. You must decide what
you consider important.
The obvious things to back up are user files
/home) and system configuration files
/etc, but possibly other things scattered
all over the filesystem).