sub depth {
    return 0 unless ref $_[0] eq 'ARRAY';
    return depth( ${$_[0]}[0] ) + 1;
}


##</code><code>##

sub depth {
    my $array = shift;
    my $count = 0;
    while (ref $array eq 'ARRAY') {
        $array = ${$array}[0];
        ++$count;
    }
    return $count;
}


##</code><code>##

Benchmark: timing 100000 iterations of loop, recursion...
      loop:  6 wallclock secs ( 5.30 usr +  0.00 sys =  5.30 CPU) @ 18867.92/s (n=100000)
 recursion:  6 wallclock secs ( 5.87 usr +  0.00 sys =  5.87 CPU) @ 17035.78/s (n=100000)


##</code><code>##

#!/usr/bin/perl

use Benchmark;
use strict;

timethese (100000, {
    'recursion' =>  q!
        sub depth {
            return 0 unless ref $_[0] eq 'ARRAY';
            return depth( ${$_[0]}[0] ) + 1;
        };

        my $array = [ [ [ 1 ], [ 1 ] ], [ [ 1 ], [ 1 ] ]];
        my $result = depth($array);
    !,
    'loop'      =>  q!
        sub depth {
            my $array = shift;
            my $count = 0;
            while (ref $array eq 'ARRAY') {
                $array = ${$array}[0];
                ++$count;
            }
            return $count;
        }

        my $array = [ [ [ 1 ], [ 1 ] ], [ [ 1 ], [ 1 ] ]];
        my $result = depth($array);
    !
});