Today's videogames are mostly all about action, story, and character building. There are very few if any good games out there today that focus on exploring unmapped territory, forging your own paths, and fresh adventure. Certainly there is plenty of great value in stirring (mindless?) action and great stories that can be told through videogames, but the thrill of adventuring in unknown lands and discovering secrets unique to your experience is priceless, and rare. Some modern games claim to do something like this, but few I've come in contact with do it in all that compelling a way. They usually lose their way while attempting to deliver their essentially conventional story.
MMORPGs sometimes offer this thrill for me, mostly because of the initial unfamiliarity with the environment and the sense of danger there oftentimes is when traveling to a new location. The first time I played Ultima Online I was dropped into a new world and with almost zero linear direction or assistance, I was off to do whatever I wanted. Whether it was adventuring with some random dudes, finding a cave or mountain pass somewhere and exploring, hunting animals or monsters in the forest and selling the hides or meat to merchants in town, it was my choice, and it was fun.
Later, for a few months, Everquest offered me some of this style of fun, it being also a new world with which I was not very familiar...but it was inferior to Ultima. The experience was more rigid, the world less malleable, and the options were more limited. This more or less defines my attitude towards the MMORPGs of today. They're expensive and not for casual consumption, and interaction with the countless other human players all doing the exact same thing as you kind of cheapens the experience.
Another common problem that gets in my way is the integration of story in these games. Most I've played give you little or no in-game story elements, making the overall experience feel stale and lifeless, as with UO. Also, most often the story that is offered is just played out before your generic avatar. He/she's entirely interchangeable with any other person on earth, making the story that much less genuine. Other, non-online RPG games have this problem, such as "Oblivion". It may be fun for a little while, but it's just so sterile, unexciting, and profitless. There has to be a balance between these two aspects I'm talking about: story and the freedom of exploration.
Some of the earliest Nintendo games were, either accidentally or intentionally, excellent examples of this balance, despite the lack of depth (come on, they were NES games...) The worlds they presented had texture, there was a basic but interesting narrative, and they even had memorable characters. Those characters, though, were just generic enough that it felt like you were the one adventuring and experiencing the story.
Zelda 1 was almost an open-world sandbox game... you could go into just about any dungeon in any order, with a few exceptions where you needed a certain item to cross a river or something. Otherwise, though, the only thing really limiting you was your skill as a player. Naturally, the more hearts you gained and items you collected, the easier it was to progress and overcome the game's obstacles, but rarely were they absolutely necessary for you to proceed.
Metroid is another example. The game is still fairly linear; there were plenty of places where you had to upgrade your suit and weaponry to progress, but the overall experience was a more organic and natural way to explore the environment and overcome challenges than is usually seen today. The main reason for this is likely the standard emphasis on story-based progression in today's videogames. We've come to expect it in most games and it seems like everyone assumes it's absolutely necessary.
When the focus is on the story, story elements must be revealed in a linear way building up to the climax, and the easiest way to present them in that order is to force the player to proceed through the game in a linear fashion. Even sandbox games like Grand Theft Auto 4 have you doing this, because they are concerned about telling you a story, not letting you create one yourself, or even letting you discover it yourself.
My ideal adventure/exploration game does not throw out the idea of offering a compelling story, but it necessitates presenting it in a different way. Having a game focus on the principles I value in such a game, while at the same time featuring a main hero with a history, genuine character development, and his/her involvement in a plot reaching a satisfying conclusion, can be tricky, to say the least, but it's most certainly doable.
This all being said, it's not like I can say this definitely isn't being done today. I've yet to play Fallout 3, for example, which seems to be aiming for something like this, and does, whatever it does, well. I did play Fable, but wasn't very impressed, so I can't say that Fable 2 looks all that promising... An interesting way of presenting a story is found in Left4Dead, with story elements being found scrawled on the walls wherever you go...
My most fervent hope for the genre is that Nintendo brings it back to the franchises that began it all for me; Zelda and Metroid. Nintendo has been focusing on "innovation" for a long time, so why not try innovating their way back to their roots? How about you come up with a way to present this long lost form of art and entertainment to today's gaming audience?
Of course, this is an unrealistic wish, so I'll probably just go off and do it myself with my own material.