Not quite an every whatever, but close. Here's a little game:
Take your basic county map of Ohio.
Starting at Adams, and ending in Wyandot, make your way through all 88 counties in alphabetical order. Obviously this is impossible to do all in one trip, but the game is to see how few "strikes" you get. Every time you have to start over (i.e. you reach a county that has no non-visited adjacent counties that are after it alphabetically), you choose an adjacent county that is before it alphabetically, and count yourself with one strike.
I'm thinking that for Ohio the "magic" number would be somewhere around 12-15.
Oddly, there do not appear to be any states that you can do this without strikes, unless you count Hawaii which has kind of "fuzzy" borders due to the water. Delaware and Rhode Island (which have the fewest counties at 3 and 5 respectively) are doable with a single strike, as is Connecticut (8 counties).
New Hampshire (10 counties) is an interesting case:
Without the rules of having to start and end at the alphabetically extreme counties (Belknap and Sullivan), it's doable with one strike (Belknap-Carroll-Coos-Grafton-Merrimack-Sullivan, Cheshire-Hillsborough-Rockingham-Strafford). But if you force those boundary conditions, it takes 3 (Belknap-Carroll-Coos-Grafton-Merrimack-Strafford, Rockingham, Hillsborough, Cheshire-Sullivan).
I like the symmetry of starting and ending in the extremes so we're doin' it baby!