Thought this one would be interesting; if there is another state with three x5 north-south Interstates and five x1 north south US routes, I do not know what it is. Alas, east-west routes tend to go through more counties than do north-south routes, and it helps to have the grid pinched by a Great Lake or two. Still, I think if this exercise were run for all of the contiguous states, Tennessee ends up in the top ten.
The approximate standings so far, counting all of the counties in the states featured:
IL 1273
TX 1160
IN 1148
OH 1089
MO 1005
TN 995
PA 879
(https://www.aaroads.com/forum/proxy.php?request=http%3A%2F%2Fwww.mob-rule.com%2Fuser-gifs%2FUSA%2Fboiler78.gif&hash=2180eae54f5d2b846537e19256df7c069dcb3cc2)
US-31 doesn't enter Cass County, MI (in fact, it barely gets within 5 miles of the county line). However, US-127 does fully enter Hillsdale County, MI (which isn't marked) for a few miles north of Hudson after running on the Hillsdale-Lenawee county line from the Ohio line to about two miles north of Hudson.
Above corrections have been made, as well as one in Indiana that boosts the count by a county.
US 63 used to cross the Memphis-Arkansas Bridge along with US 61, 64, 70, and 79. It was later decommissioned south of Turrell, Arkansas, and since has been extended south into Louisiana. If US 63 still entered Tennessee, the county count would increase by 34.
Interesting that there are no northwest diagonal routes. I thought there would have been at least one.
You erroneously mark US 51 as I-69. Those two white Kentucky counties (Carlisle and Ballard) to the left of your US 45 coloration should be labeled for US 51. Is I-69 signed anywhere in Tennessee?
Should be corrected as soon as the server updates. Also missed a couple of US 431 and US 641 counties in western Kentucky. The county count is now 1003.