Interstates that closely approach a state without entering it

[OCGuy81]

Interstate 10 comes pretty close to the Georgia state line (maybe about 6 miles according to the measuring tool in Google maps) to the west of Tallahassee.

[cpzilliacus]

I-70 comes within about a mile of Morgan County, West Virginia (by way of U.S. 522) at Hancock, Maryland. 

At the I-81 interchange at Halfway, Maryland (Hagerstown), I-70 is less than 4 miles from Berkeley County, West Virginia.

About 170 miles west of Hancock (by way of Breezewood, I-70/I-76 on the Pennsylvania Turnpike and New Stanton), I-70 pays a brief visit (about 14 centerline miles) to West Virginia by crossing Ohio County and Wheeling in the Northern Panhandle.

[Skye]

I-75 has a few. Alabama (Chattanooga, TN) and Indiana (Cincinnati).

[Roadgeekteen]

I-485 and South Carolina.

[gr8daynegb]

Technically, I-41 and IL if you go by the signage.

I-68 comes within a couple miles of the PA border a few times. 

I-87 comes within a mile of NJ at Exit 15 (I-287/NY 17).

I-80 ends about 3 miles west of the GWB and the NY border (qualifies because it doesn't run parallel)

For now, I-11 ends at the AZ state line. Also, I-26 and the eastern I-74 end at the VA state line. 

I-65 gets within a few miles of FL near its southern terminus

I-287 at its east end is less than a mile from the CT border (the 1 mile advanced BGS on I-95 South is near the 0.4 MP in CT).  And I-495 in DE ends just feet south of the PA border.

I've scene signage in Illinois, but it's obvious Wisconsin cares more about I-41 than Illinois does.

I remember the pre internet days where I-39 and Wisconsin was on such a list.......and then they finished the 78/90/94 interchange(I remember that exit being a stop sign exit to go left or right onto WI-78)

[Henry]

I-294 almost makes it to IN, but it just ends concurrent with I-80 on the IL side, where I-94 joins to cross the border afterwards.