What is the only U.S. state that does not have an interstate highway?

Alaska



Both Alaska and Puerto Rico have public roads that receive funding from the Interstate program, although these routes are not signed as Interstate Highways (except on paper). These roads are neither planned for, nor built to, official Interstate Highway standards


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

Alaska. It has no interstate highways.
https://en.wikipedia.org/wiki/Alaska

Alaska is the only one.

Here's a casual, highly readable explanation:
http://www.fhwa.dot.gov/infrastructure/hawaii.cfm

This one offers more depth:
http://geography.about.com/od/urbaneconomicgeography/a/interstates.htm

Key statement:
"Although Alaska has no Interstate Highways, Hawaii does. On the island of Oahu are the Interstates H1, H2, and H3 which connect important military facilities on the island."

This despite the Alcan Highway that connects Alaska to the Lower 48! It's more about definitions than logic.
https://en.wikipedia.org/wiki/Alaska_Highway