RPG-ish boardgames are my favorite type of game these days when I'm not prepared to play D&D, D20 Modern, or M&M 2. My favorites include:
Descent: Great game. However, it doesn't come with any campaign rules; it's meant to be played as an Overlord vs. the players kind of game. There's an expansion due out in the winter this year (likely after the first of next year knowing FFG's slow release schedule) that will include rules for campaign play.
Runebound: This is a better Talisman IMO. It has a lot going for it with two full box expansions (Isle of Dread and Midnight), and LOTS of cheap deck expansions. Runebound has a lot of replayability. Make sure you get the Second Edition. The first edition wasn't very good at all.
Arkham Horror: Yet another FFG game. Your characters advance as you travel through the city of Arkham closing gates to elder dimensions and fighting all sorts of horrors. There's an expansion for it coming out soon.
Warhammer Quest: This is hands down the best thing that GW has ever put out to date, and is far and away my favorite dungeon-delving game. It's long OOP, so finding a copy is going to be tough and is likely going to cost you a ton of money, but it's well worth it.
Hero Quest and Advanced Hero Quest: Both OOP and was produced by a partnership of MB and GW. While it will cost you to get copies of both (especially AHQ), they are worth the effort to find. I don't remember how the advancement went with AHQ (I don't own a copy and only got to play it a couple times years ago), but in HQ character advancement was purely gear upgrades.
World of Warcraft: Yet another FFG game (they are quickly becoming my favorite game maker). While this is more of a war game, it does have some RPG elements in the sense of MMORPG's. It takes a while to play, but it's a very fun game.
Doom: Yep, another FFG game. This is where the system used in Descent came about. Unlike Descent, however, Doom is more easily tailored to campaign play. It has some balance issues, but unlike a lot of people over at the Boardgame Geek I don't find it all that difficult to deal with.