What's the Closest Interstate to you that you Have Never Been on Any of it?

[Beltway]

Supplementary

If by "supplementary" you mean "auxiliary", then...
This distinction matters less to me, living in Wichita, than it probably does to people in other regions.  I-135 is an almost-hundred-mile-long trunk route through farmland and is probably the third-most important highway in the state.  So it feels weird to think of it as a second-class route just because of its three-digit number.
Anyway, that was just a total aside.  Carry on, everyone...

I don't consider them to be "second class" just that in most cases they are auxiliary to mainline routes.

It used to be I-35W and it could be argued that depicted more prominence than I-135.

Same could be said for I-70S and I-270. Nothing second class about the route that connects I-70 to Washington.

I-476 (132 miles long) deserves a mainline route as it functions like an intercity route. It could be a second iteration of I-85.

[dantheman]

Mine are
2di: I-99. I think I-70 would barely be closer if the Baltimore segment had been built as planned.
3di: I-781. I had driven by the now-eastern end on US 11 a few times before it was built, but haven't been to Watertown since.

I will be moving in a couple of months. After that, I-781 will still win, but I-180 will become only a few miles further.

[jlam]

3di: I-180 WY (~40 mi)

I've since clinched both I-180 and I-225, my next-closest untraveled Interstate. My answer for this is now I-135 in Kansas at 412 miles away, followed very closely by the western I-86 at 413 miles away.

[davewiecking]

After years of careful consideration, I've concluded I've never been on I-581. Pretty sure I've never been on I-85.

[jp the roadgeek]

Having clinched I-587 in NY today, I can now confirm the closest stretch is I-195 RI/MA.  The 2DI remains the officially designated stretch of I-86 in NY east of I-81.