Is there a way to have a greedy flip flop operator?

Not that I am aware of. However there are lots of ways to achieve the same effect. Here's one:

use strict;
use warnings;

use Test::More tests => 1;

my $want = <<EOT;
start text
+ 123
+ 456
+ 789
EOT

my $out;
while (<DATA>) {
    last unless (/^(start|\+)/);
    $out .= $_
}
is ($out, $want);

__DATA__
start text
+ 123
+ 456
+ 789
some other text not starting with +
[download]

If your actual data is different then you might have to try another approach but I think this should be fairly efficient.