For a regular expression to match at all, it does not matter whether the regex engine or the regex subexpressions are greedy or nongreedy. Both will give up or match more if that's neccessary to make the whole match succeed. If you care about how the regular expression matches, greedyness becomes an issue:

for (<DATA>) {
    print "Greedy: $1/$2\n";
        if /^(a+)(a+)$/;
    print "Non-greedy: $1/$2\n";
        if /^(a+?)(a+)$/;
};
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As you allow only one way to match, due to anchors etc., greediness can barely play any role — apart from how subpatterns match. Do you even care about subpatterns?