Barenaked Ladies, I’m embarrassed to say. We were enormous fans at the time, not so much anymore.
From Melbourne FL to Milwaukee to see The Promise Ring, they only did a handful of dates and the furthest south they got was DC.
Next week I am driving from Melbourne to Miami to see Coheed and Cambria, which is only a couple of hours, but the crazy thing is that I am seeing them the previous day in Orlando as well! When I bought the tickets I thought they were spaced out over a couple more days but oh well.
We travelled from London to NY to see Van Halen, that was so completely worth the trip! Also went from Manchester to Las Vegas to see Steel Panther.
Other than that, I live in the UK and regularly go to venues in Europe to see the bands I like - given that I can be in Paris within five hours or Brussels in seven, it’s not that much of a trek.
I’m not a big concert-goer, but I went from Saskatoon to Edmonton to see The Rolling Stones (Voodoo Lounge tour). That’s a 5 hour drive each way.
Seattle to Philadelphia in 1985 to see the Marc Chagall exhibition at the Philadelphia Museum of Art.
I live in the SF Bay Area, and I’ve gone to L.A. and to Chicago to see Gaelic Storm.
Technically, the Chicago trip was for my brother-in-law’s graduation from Navy training, but if it weren’t for the concert I might not have gone.
I’ve driven 500 miles a several times to see The Who (from San Jose to LA/San Diego).
This doesn’t really count, but I when was planning a road trip to the East Coast, I scheduled my visit to Washington DC to coincide with a Todd Rundgren show.
I took a bus by myself from Toronto to Cleveland to make my lifelong dream come true and see Duran Duran in concert.
Saw them again in Atlantic City a couple of years later - took a bus by myself again.
I took a train by myself from Toronto to Montreal to see Depeche Mode.
My husband and I flew to Baltimore to see The Gaslight Anthem. I chose that show because I’d been to Baltimore before and knew how easy it is to get from the airport to the Inner Harbour, and if I had to go alone I felt safe enough in the Inner Harbour. Turned out my husband was off so he came with me anyway.
I went from where I am all the way to Hickory, NC back in the summer of 2000 for a show which I want to say is like three hours away. 
God bless you always!!!
Holly
NYC to Bratislava, Slovakia, just to attend an all-night electronica party. Trip of a lifetime, and I made many Slovak friends using the international languages of dance and beer.
Not terribly far- my wife was dying to see U2 a few years back, so we drove from Austin to the new Cowboys Stadium in Dallas. About 4 hours, altogether.
We were miles from the stage, so I didn’t think it was a great investment, but my wife loved the show so I guess it was worth the money and travel time
I drove from Pittsburgh to Struthers, Ohio to see Tommy Castro. The venue was called The Cellar. I banged my head many times, as it literally is a cellar.
If a college bowl game can be considered a show, then the farthest is 1400 miles. Furthest for a concert is 300 miles.
My friend and I travelled to Sydney AU from San Francisco to see The White Stripes.
Pittsburgh to Orlando to see the Lords of Acid. And I have hopes some day of seeing the Rednex in either Denmark or Holland or somewhere like that.
How were they? Jack White recently stormed off the stage at Radio City in NYC because he didn’t like the audience or something like that. He barely explained why in a recent interview about it.
Back in the day, I drove about 450 miles to see Bob Mould. I remember being very frustrated about a couple of shows that I had just missed involving his short-lived band Sugar, so I resolved to go see him if he wandered anywhere near where I was. I just couldn’t wait! Thankfully, it wasn’t long afterward that I moved a little closer to civilization and no longer had the need to undertake such mammoth excursions for musical satisfaction.
4 hours to Rochester, NY to see Ben Folds Five.
4 hours to Cincinnati, OH to see the Squirrel Nut Zippers
3 hours to Springfield, OH to see some ex-SNZ members play and party with them.
6 hours to DC to see Ben Folds Five but they just happened to be in town the weekend I was there too. I would have gone to DC regardless.
8 hours to Chapel Hill, NC to see Ben Folds’ first solo show but that was more like a vacation to visit friends. Same with several other shows I’ve seen there.
Driving 2 hours to Pittsburgh this summer to see Belle & Sebastian.
From Columbus, Ohio to Bethel, NY, for Woodstock. Not for the distance, but because there were 6 of us in a VW Beetle . . . including a morbidly obese woman.
We live about 3 hours from St. Louis, so it is not unusual for us to take a weekend to see a concert or show. We have family there, which gives us a free place to stay. Latest concert/show was Paula Poundstone a couple of weeks ago.
We’ve also taken Amtrak for a long weekend in Chicago to see Eddie Izzard - 6 hours each way.
Longest trip was last October. NE Missouri to Austin for Leonard Cohen. If anyone is totally worth traveling almost 2000 round trip miles for, it is Leonard Cohen.