That's actually a problem I'm having right now. I need to find something to do, to improve my knowledge, understanding and experience of Perl, but I can't think of anything. I imagine there are loads of things I could do for work, but the LEA's filtering system prevents me from installing new modules from CPAN, so I can't get, say, Win32::GuiTest which would let me control application windows with the script. Effectively, that would let me automate our full software install by sending mouse-clicks and simulated keystrokes to the system.
Alas.
As a starter, do a couple of tutorials, then write a small program to print all of the prime numbers between 0 and 1000.
Edit: Actually, that might be a little hard for starting out...