As a suggestion, this is something I had trouble with when I was learning regex. Non-greedy not only matches as little as possible, but it also *doesn't backtrack*; This is sort of an example that gave me trouble until I read about backtracking in the man page.
$string = "<foo>...</foo><bar>...</bar>";
# This matches <foo></bar>, not what we want.
$string =~ /\<.*\>(.*)\</.*\>/;
# this works, i think.
$string =~ /\<.*?\>(.*?)\</.*?\>/;
Even if it doesn't work, I hope you get the idea. I'd show the exact example of what I was doing when I ran into this, but it's overly-complicated (removing parts of tags from a string, where there's a list of tags and attributes for those tags that need to be removed).
Greediness relies a lot on backtracking, so to avoid frustrating another fledgling perl coder with the same problems, it's at least worth a note wherever greediness is in issue.