Devel::Peek is very useful for finding out this sort of low-level detail about perl's variables. When using it to dump a string, you'll see two values listed at the end - "CUR" is the current length of the actual string, and "LEN" is the size of the memory area currently reserved for it.

A simple test of $string .= $nextchar shows that (with perl-5.8.0 on this platform at least) the LEN is changing in lockstep with CUR under this approach - perl is resizing the string buffer to be just big enough for the new string each time, so there is a lot of reallocing and therefore slow string copying going on.

One simple workaround is to presize the buffer, which of course will work best if you've got a good idea how big the string is likely to get. Here's a benchmark to demonstrate that:

use Benchmark;
Benchmark::cmpthese(shift, {
    a => q{
        $a[++$i] = "";
        $a[$i] .= "a" for 1 .. 10000;
    },
    b => q{
        $b[++$j] = "b" x 10000;
        $b[$j] = "";
        $b[$j] .= "b" for 1 .. 10000;
    },
});
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Benchmark: running a, b for at least 1 CPU seconds...
         a:  1 wallclock secs ( 1.03 usr +  0.01 sys =  1.04 CPU) @ 12
+4.04/s (n=129)
         b:  1 wallclock secs ( 1.08 usr +  0.01 sys =  1.09 CPU) @ 19
+1.74/s (n=209)
   Rate    a    b
a 124/s   -- -35%
b 192/s  55%   --
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Update: added "on this platform"