If a string in Perl 5 passes looks_like_number, it might as well be a number. For instance,
my $s = \"10\" + 5;
results in $s
$s
I can only think of one: when checking for truth. Strings that are equivalent to 0, but that are not "0", such as "0.0", "0 but true", "0e0", etc. all pass looks_like_number and evaluate to 0 in numeric context, but are still considered true values.
0
"0"
"0.0"
"0 but true"
"0e0"
looks_like_number