I put out a call for challenge requests and got some interesting ones, but what struck me that that many of them were most interesting for why they don't work, so I'm going to be doing counter-examples today. These are all things you might want to do in a game, but they're not necessarily good candidates for the combative model of challenges.
1. Convincing a judge that the defendant is innocent
I admit I'd be leery of trying this on in play because it's very hard to draw in the whole group. Legal arguments support tag teaming very poorly. However, you could make a larger challenge out of the investigation AND the case, especially if you were willing to go all Law & Order on it. That is to say, the investigation actions could diminish the SP of the prosecution's case to the point where the lawyer's arguments can do enough damage to the case to finish it off.
The problem with this is that it's difficult to create any real sense of urgency to this. There's potentially big stakes (especially if the player's are arguing their own case) but it's difficult to come up with ways for the investigators and lawyers to be threatened. Instead, you'd need to do something like put this on a timeline, allowing only a limited number of rounds. Mechanically, this works fine, but it overlooks one key element: If players are doing an investigation, then it's probably because they _like_ investigation, so doing it entirely as abstract rolls probably removes some of their enjoyment of figuring it out.
So while I wouldn't make a challenge out of it, I might steal a bit from the model to mix with an idea from Gumshoe, and put clues behind small challenges that generally don't fight back. That is to say, players will always be able to find the clues, so long as they look for them. If the philosophy of this approach is uncomfortable, then there's no need to use it. It's just an idea for using clues to get to other things, rather than playing to get to clues.
2. Indiana Jones Escaping from the Rolling Boulder
Fred actually did this one very effectively in play. Putting the boulder on the map and simply moving it forward was more than enough of a threat to represent the idea.
3. Research something in a library
Research is pretty boring stuff under the best of circumstances, but unless the books are jumping off the shelves to attack the players, there's not much back and forth to it. You're pretty much just marking time to get the job done. Curiously, this is one of the tasks that a vanilla skill challenge can handle quickly and discretely, provided you think letting your players fail at research isn't super lame. Which it is, if it's plot-driving.
Of course, if the books ARE attacking the characters, then you may have the most awesome library ever.
4. Survive a Plague
This is an interesting one for a couple of reasons (setting aside existing disease rules, which are actually pretty good). First and foremost, are you _really_ going to let your characters die from the plague? I mean, I guess you might if you've got some crazed ideas about realism, but would that really make a fun game for anybody?
Second, it's very passive. Unless you want to roll a lot of hand-washing and water-boiling actions, then this is not going to be a lot of fun or particularly interesting.
Both of these suggest that this might be a great _backdrop_ to something else (dealing with a dungeon while fighting off a disease is one of those ideas that's good on paper, but has been lame in every published implementation I've scene). Alternately, it might be a good problem at a larger scale (protect a town from the depredations of plague) , but on a personal level, it's just not engaging enough.
So why talk about what doesn't work? Because I think that's important to understanding a tool. See, here's the thing: I _could_ use the combative challenge to model any of those things, and when I was younger, I had a sort of geeky machismo which would want to do so just to illustrate that this particular tool "can do anything!". That was a little silly, because I wasn't smart enough to distinguish between something you can do and something that's not worth doing.
The purpose of this idea is not to create some sort of super-catchall replacement that keeps you from ever having to use another tool. It's to add another tool to your bag, one that handles a different kind of situation. Yes, you might use this method and stop using skill challenges (or vice versa), but as a GM, you'll be better prepared for play if you could use either, and can pick the tool that best fits the situation (or better yet, steal parts from both or either tool as needed).
This is one of those things that books and rules can only help so much with. They can help tell a GM how to use EVERYTHING, or how to use one thing really well, but the reality of play is going to take you somewhere between those points, and learning what not to use is a critical skill in navigating those waters.