The United Kingdom has been re-organizing its local government for the past 20 years or so, making the county (and county-equivalent) situation here quite a jumbled mess. Here, the following scheme is used:
This gives us a total of 185 second-level administrative units for the United Kingdom. It's a somewhat motely menagerie of different animals, but it's the best we can do given the evolving changes.
Another possible list is based on the historic counties or "lieutenancies", which eliminates many of the smaller metropolitan boroughs and unitary authorities. But that list has almost zero governmental reality behind it. So this list, despite its shortcomings, at least has real administrative machinery behind it.
Thanks to Peter Stone, who helped by adding many of these peaks to the database. I was also aided in my research by the Hill Bagging web site.