You could probably just piggyback off some random DNS server out there that permits public queries. I doubt that most domains are logging everything.

$ egrep "^[a-z]+$" /usr/share/dict/words|shuf|sed "s/$/.com/"|xargs -n1 host -t ns|grep "name server"|cut -d" " -f 4|awk '!seen[$0]++'|xargs -n1 host www.slashdot.org|awk '/^$/ {f=0} /has address/ {f=1} /^Name:/ {if (f) {print}}'
Name: ns2.afternic.com.
Name: ns1.bluehost.com.
Name: ns2.bluehost.com.
Name: ns-570.awsdns-07.net.
Name: ns1.sedoparking.com.
Name: ns02.cashparking.com.
Name: ns01.cashparking.com.
Name: ns1.namefind.com.
Name: ns2.namefind.com.

etc.

That'll look up the DNS server for a bunch of domains and, omitting duplicates, list all of the ones that can resolve "www.slashdot.org", which I imagine likely means that they'll also probably be willing to resolve other domains.

EDIT: Modified the above command line to randomize the order of domains it tries so that if multiple people use this, everyone doesn't just grab the same DNS server.