If you're thinking East coast and hiking, consider Mount Marcy. It's the highest peak in NY and the only place in the continental US where you can find some alpine species (they can also be found in Alaska). Here's a picture I took on June 14, 2008:
Any idea how AWESOME it is to take a break from hiking to dive down a 3-feet deep and 200 foot long section of snow in the middle of June? It's actually pretty amazingly awesome. We took the Northern trail (I think there are four different trails to climb the peak?), and about halfway up there's a plot of land near Indian Falls (you'll see it on a map of Mount Marcy) that's a perfect site for a tent. It's off the trail, elevated, and surrounded by trees so you know no one'll wander on you. The only downside about that spot is that you're technically not supposed to camp that high, but we didn't get caught when we went. It's about a 14 mile hike one-way, though, so you'll definitely need to bring those iodine water tablets since you can't possibly carry enough water with you.
If you're avid hikers, I'd definitely say it's worth the hike.