use strict;
use warnings;
use 5.012;

use Regexp::Common qw( balanced );


my $text = <<'END_OF_STRING';
{{Infobox
 text text text
 {{text text text
 text {{text text}}
 text}}
 {{{text {{text }}
    text }}}
}}
blah blah
blah blah
blah blah
{{Infobox
 text1 text1 text1 
 {{text1 text1 text1 
 text1 {{text1 text1}}
 text1}} 
 {{{text1 {{text1 }}
    text1 }}}
}}
{{Infobox one}}
END_OF_STRING



my $pattern = $RE{ balanced        }
                 { -begin => '{{'  }
                 { -end   => '}}'  };


while ($text =~ /($pattern)/gxms) {
    say $1;
    say '*' x 20;
}

--output:--
{{Infobox
 text text text
 {{text text text
 text {{text text}}
 text}}
 {{{text {{text }}
    text }}}
}}
********************
{{Infobox
 text1 text1 text1 
 {{text1 text1 text1 
 text1 {{text1 text1}}
 text1}} 
 {{{text1 {{text1 }}
    text1 }}}
}}
********************
{{Infobox one}}
********************