use Devel::Size qw( size );

my $s = '';
for (0..1024*1024) {
   printf("%2d %3d\n", length($s), size($s));
   $s .= 'a';
}

##</code><code>##

...
1048563 1048588
1048564 1048592
1048565 1048592
1048566 1048592
1048567 1048592
1048568 1048596
1048569 1048596
1048570 1048596
1048571 1048596
1048572 1048600
1048573 1048600
1048574 1048600
1048575 1048600
1048576 1048604