Maybe a backpacking trip? Isle Royale National Park
Isle Royale National Park (U.S. National Park Service). Enough room to be alone for solitude if that is what you want, but plenty of other hikers so you will only be as alone as you want. Unique adventure with wolves and moose (you would be LUCKY to see a wolf).
Or....a backwoods canoe trip in the Boundry Waters in Minnesota. Lots of outfitters to provide the gear you'll need and not too difficult to find someone to go with.
And the classic...hike part of the Appalachian Trail. The stretch that includes the 100 Mile Wilderness would be an interesting walk.
More exotic...a supported bicycle trip in the Alps or a walk about in New Zealand or a hike up to the summit of Mt Ranier.
Plenty adventure out there big and small. You just have to decide what you like and what stretches your limits. Do a bit of research. Talk to people who do what you think you want to do (they love to talk about their passion). Then commit to going and do it. You will never regret taking the risk. You will always regret not.