New Hampshire has the 4,000 footers list -- 48 peaks that are over 4,000 feet high.
If you claim to reach them all, the AMC (Appalachian Mountain Club) will send you a certificate of completion. It's a fairly popular endeavor to start; finishing takes some perseverance, although in the 50 years or so the "list" has been in existence, it's seen around 10,000 "official" finishers and probably many more who didn't bother recording their accomplishment with the AMC.
At any rate, the most remote peak is Owl's Head. Reaching it involves a bushwhack, an 18 mile hike, and eight river crossings -- and your reward is a summit thought to be at 4,001 feet and nothing else because there's absolutely no view. Just a rock in the middle of the forest. Somebody a few years ago with their GPS found a second summit at 4,005 feet, thus leading to a similar question -- how many people have actually tagged the "real" summit of Owl's Head, or various other peaks with broad summits.
I know I didn't find it -- I wandered around with my GPS and found a spot at 4,003 feet. Close enough? I'm not turning in my AMC completion certificate. As far as I can recall off-hand, Washington is the only one with a sign marking the exact geographic summit. The other 47 "you're on your own".
There's a saying that "close" only counts in horse shoes and hand grenades. Hiking / climbing to the "real" summit seems like a worth third entry on that list.