As a dare from a friend I compared my Perl solution to an AWK solution:
$time perl -MData::Dumper -ne '$n{length($_)}++; END {print Dumper(%n)}' bigfile.txt
$VAR1 = '1088';
$VAR2 = 349647;
real 0m1.326s
user 0m0.814s
sys 0m0.371s
$time awk 'length($0) > max { max=length($0) } END { print max }' bigfile.txt
1087
real 0m21.400s
user 0m18.596s
sys 0m0.455s
I prefer Perl, but I have no issue with AWK and I actually use it frequently.