Realizing the risk of this turning into a match of code golf (or is that the intention?), here's my solution using sed, copied from my answer to another SO question and shortened to 36 chars (30 of which are the actual sed expression). It expects the strings (each on a seperate line) to be supplied on standard input or in files passed as additional arguments.
sed 'N;s/^\(.*\).*\n\1.*$/\1\n\1/;D'
A script with sed in the shebang line weighs in at 45 chars:
#!/bin/sed -f
N;s/^\(.*\).*\n\1.*$/\1\n\1/;D
A test run of the script (named longestprefix), with strings supplied as a "here document":
$ ./longestprefix < interspecies
> interstelar
> interstate
> EOF
inters
$
