Depending a bit on whom eyes for your code goes, you might want to restyle ikegami's first suggestion above to something like

use strict;
use warnings;

my $path_head = _get_first_3( '/a/b/c/d/e/f' );

sub _get_first_3 {
    my ($path) = @_;

    my $path_part
        = qr{ [^/]+ }x; # whatever char except slash

    my $path_head_regex
        = qr{ ( \A                    # beginning of string
                ( / $path_part) {3} ) # /three/part/parts here
          }x;

    my ($head) = ($path =~ m{ $path_head_regex }x);

    return $head;
}
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