Let's talk about feats of exploration and achievement for the 21st century, and how they might be done. Space is obvious but not achievable without the backing of a government or billions of private dollars.
>unexplored caves
>unexplored rainforest
>unexplored portions of Siberia and Canada perhaps?
>unclimbed mountains
>deep ocean exploration
>high altitude lakes or inaccessible river/lakes/ponds in the middle of nowhere
My current favorite is Mount Siple, a prominent mountain that looks fairly easy to climb, but still hasn't been due to the nature of its location.
https://en.wikipedia.org/wiki/Mount_Siple