What are the only two US states that have no counties within their borders?

Louisiana. It is divided into parishes.

http://enlou.com/maps/lastate_map.htm
http://enlou.com/parishes/

Alaska has boroughs, not at all like NYC's, and not exactly like counties.

https://en.wikipedia.org/wiki/List_of_boroughs_and_census_areas_in_Alaska

Alaska does not use county names, but rather Boroughs (16) and geographical Census Areas (11).

http://www.graphicmaps.com/webimage/countrys/namerica/usstates/counties/ak.htm

In addition to Louisiana and Alaska having no counties, there are two states, Connecticut and Rhode Island, that are divided into geographic regions called counties, but those counties do not have functioning governments, as defined by the Census Bureau.

https://ce.naco.org/?find=true