What have I got wrong?

You mistranscribed that \d*+.
It should be \d++.
(I checked back through all of my notes and presentations, and its definitely shown as \d++ in them.)

The nested match goes wrong when you use \d*+, because when it encounters a nested  < the \d*+ successfully matches zero times.

And, because the \d*+ | (?&LIST) is itself in a non-backtracking loop: (\d*+ | (?&LIST) )*+, when the zero-length submatch causes the main match to fail, the regex engine can't backtrack into the alternative and try the (?&LIST) instead.

So the original match from the start of the string fails, and the regex engine skips down the string, trying again and again, until it finds the nested sublist, which it is able to match.

Incidentally, you could have watched this happen live and in colour via the Regexp::Debugger module. Just download it from CPAN and add it in front of your regex:

use Regexp::Debugger;
my $re=qr{(?x) (?&LIST)
        (?(DEFINE)
            (?<LIST> < (?&ITEM)  (?: , (?&ITEM))*+ > )
            (?<ITEM> \d*+ | (?&LIST) )
        )
        };
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Just started reading Perl Best Practices, it's a real gem.

Thank you, Sir.

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

use Marpa::R2;

my $dsl = << '__DSL__';
lexeme default = latm => 1

:default ::=                    action => ::first
List     ::= ('<') Items ('>')
Items    ::= Item               action => list
           | Item (',') Items   action => merge
Item     ::= List
           | digits

digits ~ [\d]+

__DSL__

sub list  { [ $_[1] ] }
sub merge { [ $_[1], @{ $_[2] } ] }

my $grammar = 'Marpa::R2::Scanless::G'->new({ source => \$dsl });
my @cases=('<1>',
           '<<1>>',
           '<1,2>',
           '<1,2,<3,4>,5,6>',
           '<<<1,2>,3>,<4,5,<6>>,<<7>,8,9>>');
for my $case (@cases) {
    my $value_ref = $grammar->parse(\$case, 'main');
    use Data::Dumper; print Dumper $$value_ref;
}
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In the regex, changing \d*+ to \d++ fixes it, although I have to admit I don't see why yet, it'll take a bit more staring at it to figure that out... Anyway: WebPerl Regex Tester Link

Personally, I'd use Regexp::Common for this kind of task:

use warnings;
use strict;
use Regexp::Common 2013030901 qw/balanced/;

my $re = qr/$RE{balanced}{-parens=>'<>'}/;
print $re, "\n";

my @cases = ( "<1>" ,"<1,2>" ,"<1,2,<3,4>,5,6>" );
for my $case (@cases) {
    print $case, " ",
        $case =~ $re ? "matches: '$1'\n" : "doesn't match\n";
}

__END__

(?^:((?:\<(?:(?>[^\<\>]+)|(?-1))*\>)))
<1> matches: '<1>'
<1,2> matches: '<1,2>'
<1,2,<3,4>,5,6> matches: '<1,2,<3,4>,5,6>'
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I should like to thank all of you for your "help" -- I say "help" because you guys did all the real work!

What I am really after is getting a regex similar to Abigail's where "begin"/"end" must be complete words ie "\bbegin\b" and "\bend\b". Note that Abigail's will accept "beginning a bend".

To the Damian, I apologize for my poor transcription ... but I worked from screen shot and *+ was the best my poor 70 year old eyesight told me.

For balanced begin/end matching, this might do:

$re = qr{
        (
            (?:
                (?&OPEN)
                (?:
                    (?>(?:.*?(?=(?&OPEN)|(?&CLOSE))))
                    |
                    (?-1)
                )*
                (?&CLOSE)
            )
        )
        (?(DEFINE)
            (?<OPEN>\bBEGIN\b)
            (?<CLOSE>\bEND\b)
        )}x;
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You're my hero!

Now to figure out exactly how it works!

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11112811
use warnings;

my $input = <<ENDOFSTRING;
beginning a bend
before stuff
begin
  one
  begin
    three
  end
  five
end
after stuff
beginning a bend
ENDOFSTRING

print $input, '-' x 70, "\n";

my $level = 0;
my $nested = $input =~ s/(\bbegin\b)|(\bend\b)|./
  ( $1 && $level++, $level ? $& : '', $2 && $level--)[1] /gesr;

print "$nested\n";
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Outputs:

beginning a bend
before stuff
begin
  one
  begin
    three
  end
  five
end
after stuff
beginning a bend
----------------------------------------------------------------------
begin
  one
  begin
    three
  end
  five
end
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