If we define a JAPH strictly as 'emit the string "Just another Perl hacker\n"' or something similarly precise (being that I'm too lazy to look up the canonical form because it's not critical to my proof), then there are an infinite number of ways to code a JAPH in Perl. Whether that infinity is countable or not I do not purpose to address.
Proof by induction:
Let the initial program P be: print "Just another Perl hacker\n"; produces the required output.
If we have a program Q to which we append:
the program's behavior will be unchanged.1;
Therefore, for any program R that is composed of P followed by any number of Q, the program RQ continues to produce the desired output.
This produces a nominally infinite sequence of programs that code a JAPH in Perl, limited only by mere physical limitations of the computer that is hosting this exercise.
QED (with caveats)