The Longest Journey and Dreamfall: The Longest Journey. I haven't played them for a while (I really should, but I'm waiting for Dreamfall: Chapters to be almost-released) but I distinctly remember them having the best storytelling I can recall in any game I've played. It's got voice acting and whatnot but even that was done well. It's no shock that the producer/writer/director of these games, Ragnar Tørnquist, is highly regarded in the gaming industry as an author as well as a game designer. He went to Oxford and has studied art, history, English and philosophy which I think has benefited him greatly.

Evidence!
Quote Originally Posted by Wikipedia: Reception of The Longest Journey
The Longest Journey was well received by the gaming press. GameSpot called it "one of the best adventure games in years" and applauded the "complex and interesting story" although found the ending lacking as "the epilogue does little to wrap everything up". IGN said the game "actually reinvents how stories can be told in the medium" and noticed the mature content, including "harsh subject matter, and some big time swearing". Some of the puzzles were described as "inane", but on the whole the game "hones the genre into its tightest, sharpest form yet". The US edition of PC Gamer praised the "mature and magical" story, the "sumptuous" graphics, and the game's puzzles. The only criticism levied by the magazine was that some parts of the game might be "too edgy" for younger players. The Longest Journey subsequently won the Adventure Game of the Year award by both gaming sites. By mid-2002, the game had sold 450,000 copies.
Quote Originally Posted by Wikipedia: Reception of Dreamfall: The Longest Journey
Most critics have been positive about Dreamfall, agreeing about the quality of the storyline, the graphical presentation and the voice acting. Some critics lament the shortness of the gameplay, and criticize the game's new combat and stealth elements as simplistic and unsatisfying.

GameSpy rated it as "outstanding":

Quote Originally Posted by GameSpy
The game covers a lot of metaphorical, political, and religious ground. It's a multi-threaded, complex affair that poses profound and troubling questions about the uses and misuses of faith, the limits of corporate, governmental, and religious power, and the significant difference between belief and fanaticism and knowledge and wisdom. More than that, though, it's also a rip-roaring adventure story filled with wonderful dialogue brought to life by a stellar cast of voice actors, clearly defined and supremely likeable characters, and graphic splendor that manages to make both worlds of super-science and bizarre magic incredibly believable.

It's been a long-time [sic] since The Longest Journey was released, with fans of the game wondering whether a sequel could possibly match the original. They need wonder no longer. Dreamfall is an amazing journey that propels players into a world where science, magic, art, and music combine to make a whole much greater than the sum of its parts.
GameSpot also ranked it as "great," stating "Dreamfall does not disappoint, for the most part. It exhibits the unique attention to detail and terrific presentation that made The Longest Journey so remarkable for its time."

Adventure Classic Gaming said "It is contemporary interactive fiction at its best. Its unique blend of storytelling and gameplay should appeal to a broad range of gamers beyond those who are loyal to the adventure genre.

Other critics, while not disputing the quality of the story, were disappointed that the game is primarily story-driven, and would like to have seen more gameplay. For example, IGN said:

Quote Originally Posted by IGN
While playing through, it's difficult to shake the impression that intelligent design was given a back seat to painfully simplistic fighting and sneaking sequences. The combat is, for all intents and purposes, a total joke. [...] While the gameplay has been drastically simplified, the story remains as engaging as ever. Characters engage in deep conversation, revealing all sorts of nuance and helping to strengthen this game's unique mood. Be warned, if you don't like watching lengthy conversations or if you generally disregard a game's plot, you'll be absolutely dissatisfied with Dreamfall. However, if you're in the mood for one of gaming's best and most recent narratives, by all means pick this one up. Just be aware that Dreamfall's appeal lies largely in its narrative and characters, and not in the gameplay."