I think perl's clobbering or removing backreferences somewhere, and you're trying to access \1 after perl's done something else with it.
If $1 is assigned to a temp var within dofunc, the code no longer dumps core. Below is my test, and output...

#!/usr/bin/perl -w
use strict;
my $weeping;
print doFunc( "AABBCC",0 );  
print doFunc( "AAAABBBBCCCC",0 );  

 sub doFunc {
    my( $theParam,$depth) = @_;
    print STDERR "\ndofunc($theParam)\n";
    my $re;
    $re = ${['(.+)$weeping']}[$depth]||'(.+)($weeping)';     
    print STDERR join"\n", $theParam =~ /$re/g,undef;
    $theParam =~ s/$re/&doFunc($weeping,$depth+1)/ge;
    $weeping=$1;
    ++$theParam;
    print STDERR "=$theParam\n";
    return $theParam;
}
[download]

and this gives us

dofunc(AABBCC)
=AABBCD
AABBCD
dofunc(AAAABBBBCCCC)
=AAAABBBBCCCD
AAAABBBBCCCD
[download]

Now, if you'll excuse me, I need to go wash my hands compulsively :-)
update so what output do you expect, out of curiosity?