Northampton County, NC, last voted Republican in 1896, so it's possible that there are similar streaks there depending on demographic change.
There are several counties in South Texas which have never voted Republican in a presidential election, so there's a decent shot of cities in that region possessing this distinction.
That part of South Texas was once heavily controlled by Republican machines, and the counties that those two were formed from appear to have been competitive in the WJB era, so depending on settlement time and patterns this might not be the case.